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Article

A Statistical Analysis of Cryptocurrencies

Stephen Chan 1, Jeffrey Chu 1, Saralees Nadarajah 1,* and Joerg Osterrieder 2

1School of Mathematics, University of Manchester, Manchester M13 9PL, UK;

stephen.chan@manchester.ac.uk (S.C.); jeffrey.chu@manchester.ac.uk (J.C.)

2School of Engineering, Zurich University of Applied Sciences, 8401 Winterthur, Switzerland;

joerg.osterrieder@zhaw.ch

*Correspondence: saralees.nadarjah@manchester.ac.uk; Tel.: +44-161-275-5912

Academic Editor: Charles S. Tapiero

Received: 7 April 2017; Accepted: 27 May 2017; Published: 31 May 2017

Abstract:

We analyze statistical properties of the largest cryptocurrencies (determined by market

capitalization), of which Bitcoin is the most prominent example. We characterize their exchange

rates versus the U.S. Dollar by ﬁtting parametric distributions to them. It is shown that returns

are clearly non-normal, however, no single distribution ﬁts well jointly to all the cryptocurrencies

analysed. We ﬁnd that for the most popular currencies, such as Bitcoin and Litecoin, the generalized

hyperbolic distribution gives the best ﬁt, while for the smaller cryptocurrencies the normal inverse

Gaussian distribution, generalized

t

distribution, and Laplace distribution give good ﬁts. The results

are important for investment and risk management purposes.

Keywords: exchange rate; distributions; blockchain; Bitcoin

JEL Classiﬁcation: C1

1. Introduction

Bitcoin, the ﬁrst decentralized cryptocurrency, has gained a large following from the media,

academics and the ﬁnance industry since its inception in 2009. Built upon blockchain technology, it has

established itself as the leader of cryptocurrencies and shows no signs of slowing down. Instead of

being based on traditional trust, the currency is based on cryptographic proof which provides many

advantages over traditional payment methods (such as Visa and Mastercard) including high liquidity,

lower transaction costs, and anonymity, to name just a few.

Indeed, the global interest in Bitcoin has spiked once again in recent months, for example, the UK

government is considering paying out research grants in Bitcoin; an increasing number of IT companies

are stockpiling Bitcoin to defend against ransomware; growing numbers in China are buying into

Bitcoin and seeing it as an investment opportunity. Perhaps most signiﬁcantly, the Chair of the Board of

Governors of the U.S. Federal Reserve has been encouraging central bankers to study new innovations

in the ﬁnancial industry. In particular, the Chair expressed a need to learn more about ﬁnancial

innovations including Bitcoin, Blockchain, and distributed ledger technologies. With this recent surge

in interest, we believe that now is the time to start studying Bitcoin (and other major cryptocurrencies)

as key pieces of ﬁnancial technology.

Since 2009, numerous cryptocurrencies have been developed, with, as of February 2017, 720 in

existence. Bitcoin is the largest and most popular representing over 81% of the total market of

cryptocurrencies (CoinMarketCap 2017). However, as the statistics show, many have not garnered the

same level of interest. The combined market capitalization of all cryptocurrencies is approximately

USD $19 billion (as of February 2017), with the top 15 currencies representing over 97% of the market,

and seven of these accounting for 90% of the total market capitalization. In our analysis, we focus on

J. Risk Financial Manag. 2017,10, 12; doi:10.3390/jrfm10020012 www.mdpi.com/journal/jrfm

J. Risk Financial Manag. 2017,10, 12 2 of 23

these seven cryptocurrencies which fall into the category of having existed for more than two years

and are within the top 15 currencies by market capitalization. These are Bitcoin, Ripple, Litecoin,

Monero, Dash, MaidSafeCoin, and Dogecoin.

There exists much research on Bitcoin—the most popular cryptocurrency. We brieﬂy discuss the

analysis and results of some of the most popular studies. Hencic and Gourieroux (2014) model and

predict the exchange rate of Bitcoin versus the U.S. Dollar, using a noncausal autoregressive process

with Cauchy errors. Their results show that the daily Bitcoin/USD exchange rate shows local trends

which could indicate periods of speculative behaviour from online trading.

Sapuric and Kokkinaki (2014)

investigate the volatility of Bitcoin, using data from July 2010 to April 2014, by comparing it to the

volatility of the exchange rates of major global currencies. Their analysis indicates that the exchange rate

of Bitcoin has high annualised volatility, however, it can be considered more stable when transaction

volume is taken into consideration.

Briere et al. (2015)

use weekly data from 2010 to 2013 to analyse

diversiﬁed investment portfolios and ﬁnd that Bitcoin is extremely volatile and shows large average

returns. Perhaps surprisingly, the results indicate that Bitcoin offers little correlation with other assets,

although it can help to diversify investment portfolios.

In Kristoufek (2015)

, the inﬂuencing factors of

the price of Bitcoin are investigated and applied to the Chinese Bitcoin market. Short and long term

links are found, and Bitcoin is shown to exhibit the properties of both standard ﬁnancial assets but also

speculative assets, which fuel further discussion on whether Bitcoin should be classed as a currency,

asset or an investment vehicle. Chu et al. (2015) give the ﬁrst statistical analysis of the exchange rate

of Bitcoin. They ﬁt ﬁfteen of the most common distributions used in ﬁnance to the log returns of

the exchange rate of Bitcoin versus the U.S. Dollar. Using data from 2011 to 2014 they show that the

generalized hyperbolic distribution gives the best ﬁt.

There are thousands of papers published on exchange rates. Hence, it is impossible provide

a review of all papers. Here, we mention seven of the most recent papers: Corlu and Corlu (2015)

compare the performance of the generalized lambda distribution against other ﬂexible distributions

such as the skewed

t

distribution, unbounded Johnson family of distributions, and the normal inverse

Gaussian distribution, in capturing the skewness and peakedness of the returns of exchange rates.

They conclude that for the Value-at-Risk and Expected Shortfall, the generalised lambda distribution

gives a similar performance, and in general it can be used as an alternative for ﬁtting the heavy

tail behaviour in ﬁnancial data; Nadarajah et al. (2015) revisit the study of exchange rate returns in

Corlu and Corlu (2015)

, and show that the Student’s

t

distribution can give a similar performance to

those of the distributions tested in Corlu and Corlu (2015); Bruneau and Moran (2017) investigate

the effect of exchange rate ﬂuctuations on labour market adjustments in Canadian manufacturing

industries; Dai et al. (2017) examine the role of exchange rates on economic growth in east Asian

countries; Parlapiano et al. (2017) examine exchange rate risk exposure on the value of European ﬁrms;

Schroeder (2017) investigates the macroeconomic performance in developing countries with respect to

exchange rates; Seyyedi (2017) provides an analysis of the interactive linkages between gold prices, oil

prices, and the exchange rate in India.

One of the aspects we are looking at is the volatility of cryptocurrencies. There are numerous

deﬁnitions and methods for computing volatility in the literature. For example, in Sapuric and Kokkinaki

(2014) volatility is defined as the annualised volatility (or the standard deviation) representing the daily

volatility of an exchange rate. This is calculated by multiplying the standard deviation of the exchange

rate by the square root of the number of trading days per year.

Briere et al. (2015)

use annualised returns

in their analysis and also compute the annualised volatility. Other types of volatility in the literature can

be broadly categorised as future, historical, forecast, and implied volatility (Natenberg 2007). In the

remainder of this paper, the term volatility is deﬁned as the spread of the daily exchange rates and log

returns of the exchange rates of the cryptocurrencies over the time period considered.

The paper is organised as follows. In Section 2, we give an overview of the data used in our

analysis including descriptions and sources. Section 3examines the statistical properties of the

cryptocurrencies by ﬁtting a wide range of parametric distributions to the data. Section 4provides

J. Risk Financial Manag. 2017,10, 12 3 of 23

a discussion of our results. Finally, Section 5concludes and summarizes our ﬁndings. Throughout the

paper, 0.000 should not be interpreted as an absolute zero. It only means that the ﬁrst three decimal

places are equal to zero.

2. Data

The data used in this paper are the historical global price indices of cryptocurrencies, and

were obtained from the BNC2 database from Quandl. The global price indices were used as they

represent a weighted average of the price of the respective cryptocurrencies using prices from

multiple exchanges. For our analysis, we choose to use daily data from 23 June 2014 until the

end of February 2017. A start date of June 2014 was deliberately chosen so that we can analyze seven

of the top ﬁfteen cryptocurrencies, ranked by their market capitalization, as of February 2017—see

CoinMarketCap (2017)

for the current rankings of cryptocurrencies by market capitalization. The seven

cryptocurrencies chosen to be part of our analysis are: Bitcoin, Dash, LiteCoin, MaidSafeCoin, Monero,

DogeCoin, and Ripple. It should be noted that we omitted Ethereum, arguably the second largest

cryptocurrency at present, and Ethereum Classic, as those two only started trading in 2015 and 2016,

respectively. Other notable cryptocurrencies such as Agur and NEM were also omitted due to the lack

of data. We believe that our choice of cryptocurrencies covers the most prominent currencies, and

indeed they represent 90% of the market capitalization as of February 2017 (CoinMarketCap 2017).

In the following, we provide a brief introduction of the seven cryptocurrencies chosen.

Bitcoin

is undoubtedly the most popular and prominent cryptocurrency. It was the ﬁrst realisation

of the idea of a new type of money, mentioned over two decades ago, that “uses cryptography

to control its creation and transactions, rather than a central authority” (Bitcoin Project 2017).

This decentralization means that the Bitcoin network is controlled and owned by all of its users,

and as all users must adhere to the same set of rules, there is a great incentive to maintain the

decentralized nature of the network. Bitcoin uses blockchain technology, which keeps a record of

every single transaction, and the processing and authentication of transactions are carried out by the

network of users (Bitcoin Project 2017). Although the decentralized nature offers many advantages,

such as being free from government control and regulation, critics often argue that apart from its users,

there is nobody overlooking the whole system and that the value of Bitcoin is unfounded. In return for

contributing their computing power to the network to carry out some of the tasks mentioned above,

also known as “mining”, users are rewarded with Bitcoins. These properties set Bitcoin apart from

traditional currencies, which are controlled and backed by a central bank or governing body.

Dash

(formerly known as Darkcoin and XCoin) is a “privacy-centric digital currency with instant

transactions” (The Dash Network 2017). Although it is based upon Bitcoin’s foundations and shares

similar properties, Dash’s network is two-tiered, improving upon that of Bitcoin’s. In contrast

with Bitcoin, Dash is overseen by a decentralized network of servers—known as “Masternodes”

(The Dash Network 2017)

which alleviates the need for a third party governing body, and allows for

functions such as ﬁnancial privacy and instant transactions. On the other hand, users or “miners” in the

network provide the computing power for basic functions such as sending and receiving currency, and

the prevention of double spending. The advantage of utilizing Masternodes is that transactions can be

conﬁrmed almost in real time (compared with the Bitcoin network) because Masternodes are separate

from miners, and the two have non-overlapping functions (The Dash Network 2017). Dash utilizes the

X11 chained proof-of-work hashing algorithm which helps to distribute the processing evenly across

the network while maintaining a similar coin distribution to Bitcoin. Using eleven different hashes

increases security and reduces the uncertainty in Dash. Dash operates “Decentralized Governance

by Blockchain” (The Dash Network 2017) which allows owners of Masternodes to make decisions,

and provides a method for the platform to fund its own development.

LiteCoin (LTC)

was created in 2011 by Charles Lee with support from the Bitcoin community.

Based on the same peer-to-peer protocol used by Bitcoin, it is often cited as Bitcoin’s leading rival

as it features improvements over the current implementation of Bitcoin. It has two main features

J. Risk Financial Manag. 2017,10, 12 4 of 23

which distinguish it from Bitcoin, its use of scrypt as a proof-of-work algorithm and a signiﬁcantly

faster conﬁrmation time for transactions. The former enables standard computational hardware to

verify transactions and reduces the incentive to use specially designed hardware, while the latter

reduces transaction conﬁrmation times to minutes rather than hours and is particularly attractive in

time-critical situations (LiteCoin Project 2017).

MaidSafeCoin

is a digital currency which powers the peer-to-peer Secure Access For Everyone

(SAFE) network, which combines the computing power of all its users, and can be thought of as

a “crowd-sourced internet” (MaidSafe 2017a). Each MadeSafe coin has a unique identity and there

exists a hard upper limit of 4.3 billion coins as opposed to Bitcoin’s 21 million. As the currency is

used to pay for services on the SAFE network, the currency will be recycled meaning that in theory

the amount of MaidSafe coins will never be exhausted. The process of generating new currency

is similar to other cryptocurrencies and in the case of the SAFE network it is known as “farming”

(MaidSafe 2017b). Users contribute their computing power and storage space to the network and are

rewarded with coins when the network accesses data from their store (MaidSafe 2017b).

Monero (XMR)

is a “secure, private, untraceable currency” (Monero 2017) centred around

decentralization and scalability that was launched in April 2014. The currency itself is completely

donation-based, community driven and based entirely on proof-of-work. Whilst transactions in the

network are private by default, users can set their level of privacy allowing as much or as little

access to their transactions as they wish. Although it employs a proof-of-work algorithm, Monero

is more similar to LiteCoin in that mining of the currency can be done by any modern computer

and is not restricted to specially designed hardware. It arguably holds some advantages over other

Bitcoin-based cryptocurrencies such as having a dynamic block size (overcoming the problem of

scalability), and being a disinﬂationary currency meaning that there will always exist an incentive to

produce the Monero currency (Monero 2017).

Dogecoin

(Dogecoin 2017) originated from a popular internet meme in December 2013. Created

by an Australian brand and marketing specialist, and a programmer in Portland, Oregon, it initially

started off as a joke currency but quickly gained traction. It is a variation on Litecoin, running on the

cryptographic scrypt enabling similar advantages over Bitcoin such as faster transaction processing

times. Part of the attraction of Dogecoin is its light-hearted culture and lower barriers to entry to

investing in or acquiring cryptocurrencies. One of the most popular uses for Dogecoin is the tipping of

others on the internet who create or share interesting content, and can be thought of as the next level

up from a “like” on social media or an “upvote” on internet forums. This in part has arisen from the

fact that it has now become too expensive to tip using Bitcoin.

Ripple

was originally developed in 2012 and is the ﬁrst global real-time gross settlement network

(RTGS) which “enables banks to send real-time international payments across networks” (Ripple 2017).

The Ripple network is a blockchain network which incorporates a payment system, and a currency

system known as XRP which is not based on proof-of-work like Monero and Dash. A unique property

of Ripple is that XRP is not compulsory for transactions on the network, although it is encouraged

as a bridge currency for more competitive cross border payments (Ripple 2017). The Ripple protocol

is currently used by companies such as UBS, Santander, and Standard Chartered, and increasingly

being used by the ﬁnancial services industry as technology in settlements. Compared with Bitcoin,

it has advantages such as greater control over the system as it is not subject to the price volatility of the

underlying currencies, and it has a more secure distributed authentication process.

Summary statistics of the exchange rates and log returns of the exchange rates of the seven

cryptocurrencies are given in Tables 1and 2, respectively. In Table 1, the summary statistics for the

raw exchange rates of the cryptocurrencies, we see a simple reﬂection of the “worth” or value of each

currency. It can clearly be seen that the exchange rate of Dogecoin is the least signiﬁcant, and at the

time of writing (February 2017), the exchange rate is approximately $0.0002 USD to one Dogecoin.

This supports the evidence that Dogecoin is primarily used as a currency for online tipping, rather

than as a currency for standard payments. It has the lowest minimum, ﬁrst quartile, median, mean,

J. Risk Financial Manag. 2017,10, 12 5 of 23

third quartile, and maximum values. In contrast, being the most popular cryptocurrency, Bitcoin

has the largest minimum, ﬁrst quartile, median, mean, third quartile, and maximum values, which

show its greater signiﬁcance and higher “value” to those with a vested interest in cryptocurrencies.

The exchange rates of all seven currencies are positively skewed, with Litecoin, Monero, and Ripple

being the most skewed. In terms of kurtosis, MaidSafeCoin shows less peakedness than that of the

normal distribution; Bitcoin, Dash, and Dogecoin show levels similar to the normal distribution;

Litecoin, Monero, and Ripple have signiﬁcantly greater peakedness than the normal distribution.

The exchange rates of Dogecoin, MaidSafeCoin, and Ripple have the smallest variances and standard

deviations, indicating that their low volatility can perhaps be explained by the low values of the

exchange rates coupled with the fact that their range and interquartile ranges are very limited.

On the other hand, Bitcoin, Dash, and Litecoin’s exchange rates show the greatest variance and

standard deviation.

Table 1.

Summary statistics of daily exchange rates of Bitcoin, Dash, Dogecoin, Litecoin, MaidSafeCoin,

Monero, Ripple and Euro, versus the U.S. Dollar from 23 June 2014 until 28 February 2017.

Statistics Bitcoin Dash Dogecoin Litecoin MaidSafeCoin Monero Ripple Euro

Minimum 192.700 1.178 0.000 1.269 0.012 0.235 0.003 0.626

Q1 273.600 2.577 0.000 3.091 0.020 0.491 0.006 0.736

Median 415.200 3.623 0.000 3.662 0.029 0.811 0.007 0.779

Mean 447.400 5.385 0.000 3.659 0.046 2.355 0.008 0.830

Q3 593.000 7.921 0.000 4.021 0.074 1.970 0.008 0.856

Maximum 1140.000 17.560 0.000 9.793 0.152 17.590 0.028 1.207

Skewness 0.841 1.006 0.417 1.363 0.849 2.108 2.543 1.127

Kurtosis 3.096 2.992 3.175 6.621 2.503 6.526 10.693 3.067

SD 193.241 3.583 0.000 1.433 0.032 3.397 0.004 0.142

Variance 37,342.159 12.838 0.000 2.053 0.001 11.543 0.000 0.020

CV 0.432 0.665 0.294 0.392 0.695 1.443 0.471 0.171

Range 946.938 16.385 0.000 8.524 0.140 17.358 0.025 0.581

IQR 319.400 5.344 0.000 0.930 0.054 1.479 0.002 0.119

Table 2.

Summary statistics of daily log returns of the exchange rates of Bitcoin, Dash, Dogecoin,

Litecoin, MaidSafeCoin, Monero, Ripple and the Euro, versus the U.S. Dollar from 23 June 2014 until

28 February 2017.

Statistics Bitcoin Dash Dogecoin Litecoin MaidSafeCoin Monero Ripple Euro

Minimum −0.159 −0.580 −0.385 −0.278 −0.404 −0.560 −0.299 −0.046

Q1 −0.011 −0.019 −0.009 −0.010 −0.026 −0.026 −0.014 −0.004

Median −0.001 0.003 0.002 0.000 −0.001 0.002 0.002 0

Mean −0.001 −0.001 0.000 0.001 −0.002 −0.001 −0.000 −0.00004

Q3 0.008 0.020 0.015 0.009 0.023 0.028 0.017 0.003

Maximum 0.205 0.411 0.188 0.433 0.241 0.277 0.288 0.038

Skewness 0.758 −1.487 −2.506 0.756 −0.478 −1.414 −0.401 −0.145

Kurtosis 11.568 26.805 24.434 22.385 8.520 13.954 13.818 2.662

SD 0.028 0.051 0.042 0.042 0.054 0.062 0.046 0.006

Variance 0.001 0.003 0.002 0.002 0.003 0.004 0.002 0.00004

CV −47.976 −84.519 89.782 45.619 −21.499 −54.548 −96.585 −143.498

Range 0.364 0.991 0.573 0.711 0.645 0.837 0.587 0.085

IQR 0.019 0.039 0.023 0.019 0.049 0.054 0.030 0.007

Table 2gives the summary statistics for the log returns of the exchange rates of the seven

cryptocurrencies. Here, the log returns show some slightly different results. Dash, MaidSafeCoin,

and Monero have the lowest minimum values, while Dash and Litecoin have the largest maximums.

The means and medians of the log returns of all seven currencies are similar and almost equal to zero.

Only the log returns of Bitcoin and Litecoin are positively skewed, all others are negatively skewed

with Dogecoin being the most signiﬁcant. Log returns of all seven currencies have a peakedness

signiﬁcantly greater than that of the normal distribution, with the most peaked being those of Dash,

J. Risk Financial Manag. 2017,10, 12 6 of 23

Dogecoin, and Litecoin. Noteworthy, as much has been discussed about the volatility of Bitcoin returns,

the log returns of Bitcoin have the lowest variance and standard deviation of the seven cryptocurrencies.

Those with the highest variation are Dash, and perhaps unexpectedly, MaidSafeCoin and Monero.

Also shown in Tables 1and 2are the summary statistics of the exchange rates of the Euro and the

summary statistics of its log returns. For the exchange rates, the summary statistics for the Euro appear

much smaller than those of Bitcoin but comparable to those of other cryptocurrencies. For the log

returns of the exchange rates, the summary statistics for the Euro generally appear smaller compared

to all the cryptocurrencies. An exception is the coefﬁcient of variation. The magnitude of this statistic

for the Euro appears largest compared to all the cryptocurrencies.

Fitting of a statistical distribution usually assumes that the data are independent and

identically distributed (i.e., randomness), have no serial correlation, and have no heteroskedasticity.

We tested for randomness using the difference sign and rank tests. We tested for no serial correlation

using Durbin and Watson (1950,1951,1971)’s method. We tested for no heteroskedasticity using

Breusch and Pagan (1979)’s test

. These tests showed that the log returns of the exchange rates of the

seven cryptocurrencies can be assumed to be approximately independent and identically distributed,

have no serial correlation, and have no heteroskedasticity.

3. Distributions Fitted

Having brieﬂy examined the summary statistics for both the exchange rates and the log returns of

exchange rates of the seven cryptocurrencies, we provide a visual representation of the distribution of

the log returns. Figure 1shows the histograms of the daily log returns of the exchange rate (versus the

U.S. Dollar) for all seven cryptocurrencies. From the plots, we ﬁnd that the log returns in the cases of

all seven cryptocurrencies show signiﬁcant deviation from the normal distribution. Next, we proceed

to ﬁt the parametric distributions to the data.

Bitcoin

Log returns

Frequency

−0.2 −0.1 0.0 0.1 0.2

0 300

Dash

Log returns

Frequency

−0.6 −0.2 0.0 0.2 0.4

0 300

Dogecoin

Log returns

Frequency

−0.4 −0.2 0.0 0.1 0.2

0 300

Litecoin

Log returns

Frequency

−0.2 0.0 0.2 0.4

0 300

MaidSafecoin

Log returns

Frequency

−0.4 −0.2 0.0 0.2

0 300

Monero

Log returns

Frequency

−0.6 −0.4 −0.2 0.0 0.2

0 300

Ripples

Log returns

Frequency

−0.3 −0.1 0.1 0.2 0.3

0 300

Figure 1.

Histograms of daily log returns of the exchange rates of the seven cryptocurrencies versus

the U.S. Dollar from 23 June 2014 until 28 February 2017.

J. Risk Financial Manag. 2017,10, 12 7 of 23

Let

X

denote a continuous random variable representing the log returns of the exchange rate of

the cryptocurrency of interest. Let

f(x)

denote the probability density function (pdf) of

X

. Let

F(x)

denote the cumulative distribution function (cdf) of

X

. We suppose

X

follows one of eight possible

distributions, the most popular parametric distributions used in ﬁnance. They are speciﬁed as follows:

•the Student’s tdistribution (Gosset 1908) with

f(x) = K(ν)

σ"1+(x−µ)2

σ2ν#−(1+ν)/2

for

−∞<x<∞

,

−∞<µ<∞

,

σ>

0 and

ν>

0, where

K(ν) = √νB(ν/2, 1/2)

and

B(·

,

·)

denotes the beta function deﬁned by

B(a,b) = Z1

0ta−1(1−t)b−1dt;

•the Laplace distribution (Laplace 1774) with

f(x) = 1

2σexp −|x−µ|

σ

for −∞<x<∞,−∞<µ<∞and σ>0;

•the skew tdistribution (Azzalini and Capitanio 2003) with

f(x) = K(ν)

σ"1+(x−µ)2

σ2ν#−(1+ν)/2

+2K2(ν)λ(x−µ)

σ22F1 1

2,1+ν

2;3

2;−λ2(x−µ)2

σ2ν!

for

−∞<x<∞

,

−∞<µ<∞

,

−∞<λ<∞

,

σ>

0 and

ν>

0, where

2F1(a

,

b

;

c

;

x)

denotes the

Gauss hypergeometric function deﬁned by

2F1(a,b;c;x)=

∞

∑

k=0

(a)k(b)k

(c)k

xk

k!,

where (e)k=e(e+1)··· (e+k−1)denotes the ascending factorial;

•the generalized tdistribution (McDonald and Newey 1988) with

f(x) = τ

2σν1/νB(ν, 1/τ)1+1

ν

x−µ

σ

τ−(ν+1/τ)

for −∞<x<∞,−∞<µ<∞,σ>0, ν>0 and τ>0;

•the skewed Student’s tdistribution (Zhu and Galbraith 2010) with

f(x) = K(ν)

σ

(1+1

νx−µ

2σα 2)−ν+1

2

, if x≤µ,

(1+1

νx−µ

2σ(1−α)2)−ν+1

2

, if x>µ

for −∞<x<∞,−∞<µ<∞, 0 <α<1 and ν>0;

J. Risk Financial Manag. 2017,10, 12 8 of 23

•the asymmetric Student’s tdistribution (Zhu and Galbraith 2010) with

f(x) = 1

σ

α

α∗K(ν1)(1+1

ν1x−µ

2σα∗2)−ν1+1

2

, if x≤µ,

1−α

1−α∗K(ν2)(1+1

ν2x−µ

2σ(1−α∗)2)−ν2+1

2

, if x>µ

for −∞<x<∞,−∞<µ<∞, 0 <α<1, ν1>0 and ν2>0, where

α∗=αK(ν1)

αK(ν1)+ (1−α)K(ν2);

•the normal inverse Gaussian distribution (Barndorff-Nielsen 1977) with

f(x) = (γ/δ)λα

√2πK−1/2 (δγ)exp [β(x−µ)]hδ2+ (x−µ)2i−1K−1αqδ2+ (x−µ)2

for

−∞<x<∞

,

−∞<µ<∞

,

δ>

0,

α>

0 and

β>

0, where

γ=pα2−β2

and

Kν(·)

denotes

the modiﬁed Bessel function of the second kind of order νdeﬁned by

Kν(x) =

πcsc(πν)

2[I−ν(x)−Iν(x)], if ν6∈ Z,

lim

µ→νKµ(x), if ν∈Z,

where Iν(·)denotes the modiﬁed Bessel function of the ﬁrst kind of order νdeﬁned by

Iν(x) =

∞

∑

k=0

1

Γ(k+ν+1)k!x

22k+ν,

where Γ(·)denotes the gamma function deﬁned by

Γ(a) = Z∞

0ta−1exp(−t)dt;

•the generalized hyperbolic distribution (Barndorff-Nielsen 1977) with

f(x) = (γ/δ)λα1/2−λ

√2πKλ(δγ)exp [β(x−µ)]hδ2+ (x−µ)2iλ−1/2 Kλ−1/2 αqδ2+ (x−µ)2

for −∞<x<∞,−∞<µ<∞,−∞<λ<∞,δ>0, α>0 and β>0, where γ=pα2−β2.

Many of these distributions are nested: the skew

t

distribution for

λ=

0 is the Student’s

t

distribution; the generalized

t

distribution for

τ=

2 is the Student’s

t

distribution; the skewed Student’s

t

distribution for

α=

1

/

2 is the Student’s

t

distribution; the asymmetric Student’s

t

distribution for

ν1=ν2

is the skewed Student’s

t

distribution; the generalized hyperbolic distribution for

λ=−

1

/

2 is

the normal inverse Gaussian distribution; and so on.

All but one of the distributions (the Laplace distribution) are heavy tailed. Heavy tails are common

in ﬁnancial data. The Student’s

t

distribution is perhaps the simplest of the heavy tailed distributions.

It does not allow for asymmetry. The skew

t

distribution due to Azzalini and Capitanio (2003) is

an asymmetric generalization of the Student’s

t

distribution. The generalized

t

distribution due to

McDonald and Newey (1988) has two parameters controlling its heavy tails, adding more ﬂexibility.

J. Risk Financial Manag. 2017,10, 12 9 of 23

The skewed Student’s

t

distribution due to Zhu and Galbraith (2010) is a generalization of the Student’s

t

distribution with the scale allowed to be different on the two sides of

µ

. This distribution is useful if

positive log returns have a different scale compared to negative log returns. The asymmetric Student’s

t

distribution due to Zhu and Galbraith (2010) is a generalization of the Student’s

t

distribution with

the scale as well as the tail parameter allowed to be different on the two sides of

µ

. This distribution is

useful if positive log returns also have a different heavy tail behavior compared to negative log returns.

The generalized hyperbolic distribution due to Barndorff-Nielsen (1977) accommodates semi heavy

tails. It is popular in ﬁnance because it contains several heavy tailed distributions as particular cases.

The maximum likelihood method was used to ﬁt each distribution. If

x1

,

x2

,

. . .

,

xn

is a random

sample of observed values on

X

and if

Θ=(θ1,θ2, . . . , θk)0

are parameters specifying the distribution

of Xthen the maximum likelihood estimates of Θare those maximizing the likelihood

L(Θ)=

n

∏

i=1

f(xi;Θ)

or the log likelihood

ln L(Θ)=

n

∑

i=1

ln f(xi;Θ),

where

f(·)

denotes the pdf of

X

. We shall let

b

Θ=b

θ1,b

θ2, . . . , b

θk0

denote the maximum likelihood

estimate of

Θ

. The maximization was performed using the routine optim in the R software package

(R Development Core Team 2017). The standard errors of

b

Θ

were computed by approximating the

covariance matrix of b

Θby the inverse of observed information matrix, i.e.,

cov b

Θ≈

∂2ln L

∂θ2

1

∂2ln L

∂θ1∂θ2· ·· ∂2ln L

∂θ1∂θk

∂2ln L

∂θ2∂θ1

∂2ln L

∂θ2

2··· ∂2ln L

∂θ2∂θk

.

.

..

.

.....

.

.

∂2ln L

∂θk∂θ1

∂2ln L

∂θk∂θ2·· · ∂2ln L

∂θ2

k

−1Θ=b

Θ

.

Many of the ﬁtted distributions are not nested. Discrimination among them was performed using

various criteria:

•the Akaike information criterion (Akaike 1974) deﬁned by

AIC =2k−2 ln Lb

Θ;

•the Bayesian information criterion (Schwarz 1978) deﬁned by

BIC =kln n−2 ln Lb

Θ;

•the consistent Akaike information criterion (CAIC) (Bozdogan 1987) deﬁned by

CAIC =−2 ln Lb

Θ+k(ln n+1);

•the corrected Akaike information criterion (AICc) (Hurvich and Tsai 1989) deﬁned by

AICc =AIC +2k(k+1)

n−k−1;

J. Risk Financial Manag. 2017,10, 12 10 of 23

•the Hannan-Quinn criterion (Hannan and Quinn 1979) deﬁned by

HQC =−2 ln Lb

Θ+2kln ln n.

The ﬁve discrimination criteria above, used to discriminate between and determine the best

ﬁtting distribution, all utilise the maximum likelihood estimate. In all cases, the smaller the values

of the criteria the better the ﬁt. The Akaike information criterion comprises two parts: the bias

(log likelihood) and variance (parameters) (Hu 2007), and the larger the log likelihood the better the

goodness of ﬁt. However, the criterion includes a penalty term which is dependent on the number

of parameters in the model. This penalty term increases with the number of estimated parameters

and discourages overﬁtting. The Bayesian information criterion is very similar to that of the Akaike

information criterion, the only difference being that the penalty term is not twice the number of

estimated parameters, but instead is the number of parameters multiplied by the natural logarithm of

the number of observed data points. The two criteria possess different properties and thus their use is

also dependent on different factors. In addition, the Bayesian information criterion is asymptotically

efﬁcient, while the Akaike information criterion is not (Vrieze 2012). The consistent Akaike information

criterion and the corrected Akaike information criterion are also very similar to both the Akaike and

Bayesian information criteria. The former acts as a direct extension of the Akaike information criterion

in that it is asymptotically efﬁcient and still includes a penalty term which penalises overﬁtting more

strictly (Bozdogan 1987). On the other hand, the latter includes a correction for small sample bias

and includes an additional penalty term which is a function of the sample size

(Anderson et al. 2010).

Finally, an alternative to the Akaike and Bayesian information criteria is the Hannan-Quinn information

criterion. The expression for the Hannan-Quinn criterion is the same as that for the Akiake information

criterion, however, the parameter term is multiplied by the double logarithm of the sample size.

In general, the Hannan-Quinn criterion penalises models with a greater number of parameters more

compared to both the Akaike and Bayesian information criteria. However, it tends to show signs of

overﬁtting when the sample size is small (McNelis 2005). For a more detailed discussion on these

criteria, see Burnham and Anderson (2004) and Fang (2011).

Apart from the ﬁve criteria, various other measures could be used to discriminate between

non-nested models. These could include:

•the Kolmogorov-Smirnov statistic (Kolmogorov 1933;Smirnov 1948) deﬁned by

KS =sup

x

1

n

n

∑

i=1

I{xi≤x}−b

F(x),

where I{·}denotes the indicator function and b

F(·)the maximum likelihood estimate of F(x);

•the Anderson-Darling statistic (Anderson and Darling 1954) deﬁned by

AD =−n−

n

∑

i=1nln b

Fx(i)+ln h1−b

Fx(n+1−i)io,

where x(1)≤x(2)≤ ··· ≤ x(n)are the observed data arranged in increasing order;

•the Cramer-von Mises statistic (Cramer 1928;Von Mises 1928) deﬁned by

CM =1

12n+

n

∑

i=12i−1

2n−b

Fx(i).

Once again the smaller the values of these statistics the better the ﬁt. The use of these statistics in

Section 4instead of the ﬁve criteria led to the same conclusions.

J. Risk Financial Manag. 2017,10, 12 11 of 23

4. Results

In this section, we provide our analysis in terms of the best ﬁtting distributions and the results for

the log returns of the different cryptocurrencies. The results provided are in terms of log likelihood

values, information criteria, goodness of ﬁt tests, probability plots, quantile plots, plots of two

important ﬁnancial risk measures, back-testing using Kupiec’s test and dynamic volatility.

4.1. Fitted Distributions and Results

The eight distributions in Section 3were ﬁtted to the data described in Section 2. The method

of maximum likelihood was used. The parameter estimates and their standard errors for the best

ﬁtting distributions in each case are given in Table 3. The log likelihood values, and the values of

AIC, AICc, BIC, HQC and CAIC for the ﬁtted distributions (for each of the seven cryptocurrencies)

are shown in Tables 4–10. We ﬁnd that there is no one best ﬁtting distribution jointly for all seven

cryptocurrencies. However, we ﬁnd that for Bitcoin (the most popular cryptocurrency) and its nearest

rival LiteCoin, the generalized hyperbolic distribution gives the best ﬁt. For three out of the seven:

Dash, Monero, and Ripple, the normal inverse Gaussian distribution gives the best ﬁt. For Dogecoin

and MaidSafeCoin, the best ﬁtting distributions are the generalized

t

and Laplace distributions,

respectively. The adequacy of the best ﬁtting distributions is assessed in terms of Q-Q plots, P-P plots,

the one-sample Kolmogorov-Smirnov test, the one-sample Anderson-Darling test and the one-sample

Cramer-von Mises test.

Table 3. Best ﬁtting distributions and parameter estimates, with standard errors given in brackets.

Crytptocurrency Best Fitting Distribution Parameter Estimates and Standard Errors

Bitcoin Generalized hyperbolic b

µ=−0.001 (0.000),

b

δ=0.003 (0.001),

b

α=29.644 (3.707),

b

β=0.530 (1.305),

b

λ=0.220 (0.010).

Dash Normal inverse Gaussian b

µ=0.004,

b

δ=0.025,

b

α=10.714,

b

β=−2.100.

Dogecoin Generalized t b

µ=0.002 (0.000),

b

σ=0.014 (0.002),

b

p=0.893 (0.094),

b

ν=3.768 (1.269).

Litecoin Generalized hyperbolic b

µ=0.000 (0.000),

b

δ=0.006 (0.001),

b

α=10.517 (2.021),

b

β=0.412 (0.801),

b

λ=−0.186 (0.078).

MaidSafeCoin Laplace b

µ=−0.001 (0.001),

b

σ=0.0368 (0.001).

Monero Normal inverse Gaussian b

µ=0.005,

b

δ=0.040,

b

α=11.164,

b

β=−1.705.

Ripple Normal inverse Gaussian b

µ=0.003,

b

δ=0.018,

b

α=8.729,

b

β=−1.670.

J. Risk Financial Manag. 2017,10, 12 12 of 23

Table 4.

Fitted distributions and results for daily log returns of the exchange rates of Bitcoin from

23 June 2014 until 28 February 2017.

Distribution −ln LAIC AICC BIC HQC CAIC

Student t−2303.9 −4601.8 −4601.8 −4587.3 −4596.3 −4584.3

Laplace −2289.3 −4574.7 −4574.7 −4565.0 −4571.0 −4563.0

Skew t−2304.0 −4600.0 −4599.9 −4580.5 −4592.6 −4576.5

GT −2325.7 −4643.3 −4643.3 −4623.8 −4635.9 −4619.8

SST −2304.0 −4600.0 −4599.9 −4580.5 −4592.6 −4576.5

AST −2304.3 −4598.6 −4598.6 −4574.3 −4589.4 −4569.3

NIG −2316.0 −4623.9 −4623.9 −4604.5 −4616.5 −4600.5

GH −2325.7 −4641.5 −4641.4 −4617.1 −4632.2 −4612.1

Table 5.

Fitted distributions and results for daily log returns of the exchange rates of Dash from

23 June 2014 until 28 February 2017.

Distribution −ln LAIC AICC BIC HQC CAIC

Student t−1704.8 −3403.6 −3403.6 −3389.0 −3398.1 −3386.0

Laplace −1689.4 −3374.8 −3374.7 −3365.0 −3371.1 −3363.0

Skew t−1707.4 −3406.8 −3406.8 −3387.3 −3399.4 −3383.3

GT −1708.9 −3409.7 −3409.7 −3390.3 −3402.3 −3386.3

SST −1707.5 −3407.0 −3406.9 −3387.5 −3399.6 −3383.5

AST −1707.5 −3405.1 −3405.0 −3380.8 −3395.8 −3375.8

NIG −1710.4 −3412.8 −3412.8 −3393.4 −3405.4 −3389.4

GH −1710.5 −3410.9 −3410.8 −3386.6 −3401.6 −3381.6

Table 6.

Fitted distributions and results for daily log returns of the exchange rates of Dogecoin from

23 June 2014 until 28 February 2017.

Distribution −ln LAIC AICC BIC HQC CAIC

Student t−2037.7 −4069.4 −4069.4 −4054.8 −4063.8 −4051.8

Laplace −1985.9 −3967.8 −3967.8 −3958.1 −3964.1 −3956.1

Skew t−2037.7 −4067.5 −4067.4 −4048.0 −4060.1 −4044.0

GT −2051.9 −4095.8 −4095.8 −4076.4 −4088.4 −4072.4

SST −2037.7 −4067.4 −4067.3 −4047.9 −4060.0 −4043.9

AST −2039.6 −4069.1 −4069.1 −4044.8 −4059.9 −4039.8

NIG −2048.1 −4088.1 −4088.1 −4068.7 −4080.7 −4064.7

GH −2052.2 −4094.3 −4094.2 −4070.0 −4085.0 −4065.0

Table 7.

Fitted distributions and results for daily log returns of the exchange rates of Litecoin from

23 June 2014 until 28 February 2017.

Distribution −ln LAIC AICC BIC HQC CAIC

Student t−2113.3 −4220.7 −4220.7 −4206.1 −4215.1 −4203.1

Laplace −2020.0 −4036.0 −4036.0 −4026.3 −4032.3 −4024.3

Skew t−2113.6 −4219.3 −4219.3 −4199.8 −4211.9 −4195.8

GT −2125.7 −4243.4 −4243.4 −4223.9 −4236.0 −4219.9

SST −2113.5 −4219.1 −4219.1 −4199.6 −4211.7 −4195.6

AST −2113.6 −4217.2 −4217.1 −4192.8 −4207.9 −4187.8

NIG −2123.9 −4239.9 −4239.9 −4220.4 −4232.5 −4216.4

GH −2130.6 −4251.2 −4251.1 −4226.9 −4241.9 −4221.9

J. Risk Financial Manag. 2017,10, 12 13 of 23

Table 8.

Fitted distributions and results for daily log returns of the exchange rates of MaidSafeCoin

from 23 June 2014 until 28 February 2017.

Distribution −ln LAIC AICC BIC HQC CAIC

Student t−1533.2 −3060.3 −3060.3 −3045.7 −3054.8 −3042.7

Laplace −1540.4 −3076.8 −3076.8 −3067.1 −3073.1 −3065.1

Skew t−1533.3 −3058.6 −3058.5 −3039.1 −3051.2 −3035.1

GT −1541.9 −3075.9 −3075.8 −3056.4 −3068.5 −3052.4

SST −1533.3 −3058.5 −3058.5 −3039.1 −3051.1 −3035.1

AST −1533.4 −3056.7 −3056.7 −3032.4 −3047.5 −3027.4

NIG −1539.0 −3070.0 −3070.0 −3050.6 −3062.6 −3046.6

GH −1542.4 −3074.9 −3074.8 −3050.6 −3065.6 −3045.6

Table 9.

Fitted distributions and results for daily log returns of the exchange rates of Monero from

23 June 2014 until 28 February 2017.

Distribution −ln LAIC AICC BIC HQC CAIC

Student t−1438.6 −2871.2 −2871.2 −2856.6 −2865.6 −2853.6

Laplace −1432.2 −2860.5 −2860.5 −2850.7 −2856.8 −2848.7

Skew t−1440.0 −2871.9 −2871.9 −2852.5 −2864.5 −2848.5

GT −1440.5 −2872.9 −2872.9 −2853.5 −2865.5 −2849.5

SST −1439.7 −2871.4 −2871.4 −2852.0 −2864.0 −2848.0

AST −1440.6 −2871.2 −2871.1 −2846.9 −2861.9 −2841.9

NIG −1441.9 −2875.8 −2875.7 −2856.3 −2868.4 −2852.3

GH −1442.0 −2874.0 −2874.0 −2849.7 −2864.8 −2844.7

Table 10.

Fitted distributions and results for daily log returns of the exchange rates of Ripple from

23 June 2014 until 28 February 2017.

Distribution −ln LAIC AICC BIC HQC CAIC

Student t−1867.6 −3729.1 −3729.1 −3714.5 −3723.6 −3711.5

Laplace −1831.9 −3659.9 −3659.9 −3650.2 −3656.2 −3648.2

Skew t−1869.5 −3731.1 −3731.0 −3711.6 −3723.7 −3707.6

GT −1870.6 −3733.1 −3733.1 −3713.7 −3725.7 −3709.7

SST −1869.3 −3730.5 −3730.5 −3711.1 −3723.1 −3707.1

AST −1869.7 −3729.4 −3729.4 −3705.1 −3720.2 −3700.1

NIG −1875.6 −3743.3 −3743.2 −3723.8 −3735.8 −3719.8

GH −1875.9 −3741.8 −3741.8 −3717.5 −3732.6 −3712.5

The best ﬁtting distributions show the following: log returns of the exchange rates of

MaidSafeCoin have light tails; log returns of the exchange rates of Dogecoin have heavy tails; log

returns of the exchange rates of Bitcoin, Dash, Litecoin, Monero and Ripple have semi heavy tails.

Among the last ﬁve, the asymmetry of tails as measured by

β

is negative for Dash, Monero and Ripple.

Of these the largest asymmetry is for Dash, followed by Monero and then Ripple. The asymmetry of

tails as measured by

β

is positive for Bitcoin and Litecoin. Of these the larger asymmetry is for Bitcoin.

It is surprising that a light tailed distribution (the Laplace distribution) gives the best ﬁt for

log returns of the exchange rates of MaidSafeCoin. However, many authors have found that the

Laplace distribution can provide adequate ﬁts to ﬁnancial data: Linden (2001) models stock returns

by the Laplace distribution; Linden (2005) models the realized stock return volatility by the Laplace

distribution; Aquino (2006) establishes that the Laplace distribution characterizes the price movements

for the Philippine stock market; Podobnik et al. (2009) show Laplace distribution ﬁts for the stock

growth rates for the Nasdaq Composite and the New York Stock Exchange Composite.

All of the cryptocurrencies show substantially larger volatility than the exchange rate of the

Euro. Nevertheless, we also observe substantial differences within the group of cryptocurrencies.

J. Risk Financial Manag. 2017,10, 12 14 of 23

The distribution of the tails of their returns ranges from light tailed via semi-heavy tailed to heavy tailed.

Given that traditional ﬁnancial instruments usually exhibit heavy tails, this is a slightly surprising,

but also a very important, result.

4.2. Q-Q Plots

The Q-Q plots for the best ﬁtting distribution for each of the seven cryptocurrencies are shown in

Figure 2. For Bitcoin, the best ﬁtting distribution captures the middle and lower parts of the data well

but not the upper tail. For Dash, the best ﬁtting distribution captures the middle, lower and upper

parts of the data well. For Dogecoin, the best ﬁtting distribution captures the middle part of the data

well but not the lower or upper tails. For Litecoin, the best ﬁtting distribution captures the middle,

lower and upper parts of the data well. For MaidSafeCoin, the best ﬁtting distribution captures the

middle and upper parts of the data well but not the lower tail. For Monero, the best ﬁtting distribution

captures the middle and upper parts of the data well but not the lower tail. For Ripple, the best ﬁtting

distribution captures the middle part of the data well but not the lower or upper tails.

−0.1 0.0 0.1 0.2

−0.1 0.0 0.1 0.2

Observed

Expected

−0.6 −0.4 −0.2 0.0 0.2 0.4

−0.6 −0.4 −0.2 0.0 0.2 0.4

Observed

Expected

−0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3

−0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3

Observed

Expected

−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4

−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4

Observed

Expected

−0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2

−0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2

Observed

Expected

−0.4 −0.2 0.0 0.2

−0.4 −0.2 0.0 0.2

Observed

Expected

Figure 2. Cont.

J. Risk Financial Manag. 2017,10, 12 15 of 23

−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3

−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3

Observed

Expected

Figure 2.

The Q-Q plots of the best ﬁtting distributions for daily log returns of the exchange rates of

Bitcoin (ﬁrst row, left), Dash (ﬁrst row, right), Dogecoin (second row, left), Litecoin (second row, right),

MaidSafeCoin (third row, left), Monero (third row, right) and Ripple (last row) from 23 June 2014 until

28 February 2017.

4.3. P-P Plots

The P-P plots for the best ﬁtting distribution for each of the seven cryptocurrencies are shown in

Figure 3. For each cryptocurrency, the best ﬁtting distribution captures the middle, lower and upper

parts of the data well.

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Observed

Expected

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Observed

Expected

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Observed

Expected

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Observed

Expected

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Observed

Expected

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Observed

Expected

Figure 3. Cont.

J. Risk Financial Manag. 2017,10, 12 16 of 23

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Observed

Expected

Figure 3.

The P-P plots of the best ﬁtting distributions for daily log returns of the exchange rates of

Bitcoin (ﬁrst row, left), Dash (ﬁrst row, right), Dogecoin (second row, left), Litecoin (second row, right),

MaidSafeCoin (third row, left), Monero (third row, right) and Ripple (last row) from 23 June 2014 until

28 February 2017.

4.4. Goodness of Fit Tests

The

p

-values of the one-sample Kolmogorov-Smirnov test for the best ﬁtting distributions

listed in Table 3are 0.095, 0.124, 0.164, 0.094, 0.119, 0.162 and 0.056. The corresponding

p

-values

of the one-sample Anderson-Darling test are 0.103, 0.144, 0.154, 0.120, 0.051, 0.157 and 0.176.

The corresponding

p

-values of the one-sample Cramer-von Mises test are 0.082, 0.116, 0.196, 0.085,

0.168, 0.088 and 0.122. Hence, all of the best ﬁtting distributions are adequate at the ﬁve percent

signiﬁcance level.

4.5. VaR and ES Plots

Value at risk (VaR) and expected shortfall (ES) are the two most popular ﬁnancial risk measures

(Kinateder 2015,2016). If

b

F(·)

denotes the cdf of the best ﬁtting distribution then VaR and ES

corresponding to probability qcan be deﬁned by

VaR(q) = b

F−1(q)

and

ES(q) = 1

qZq

0VaR(u)du,

respectively, for 0

<q<

1. Plots of VaR

(q)

and ES

(q)

for the best ﬁtting distributions for the seven

cryptocurrencies are shown in Figures 4and 5. Also shown in the ﬁgures are estimates of VaR

(q)

and ES

(q)

for daily log returns of the Euro computed using the same best ﬁtting distributions. It is

clear that each cryptocurrency is riskier than the Euro. With respect to the upper tail of VaR, Litecoin,

MainSafecoin and Monero appear to have the largest risks. With respect to the lower tail of VaR,

Monero appears to have the largest risk. Monero also has the largest risk with respect to the lower

tail of ES.

J. Risk Financial Manag. 2017,10, 12 17 of 23

0.0 0.2 0.4 0.6 0.8 1.0

−0.05 0.00 0.05

q

VaR (q)

0.0 0.2 0.4 0.6 0.8 1.0

−0.05 0.00 0.05

Bitcoin

Euro

0.0 0.2 0.4 0.6 0.8 1.0

−0.15 −0.10 −0.05 0.00 0.05 0.10

q

VaR (q)

0.0 0.2 0.4 0.6 0.8 1.0

−0.15 −0.10 −0.05 0.00 0.05 0.10

Dash

Euro

0.0 0.2 0.4 0.6 0.8 1.0

−0.10 −0.05 0.00 0.05 0.10

q

VaR (q)

0.0 0.2 0.4 0.6 0.8 1.0

−0.10 −0.05 0.00 0.05 0.10

Dogecoin

Euro

0.0 0.2 0.4 0.6 0.8 1.0

−0.10 −0.05 0.00 0.05 0.10 0.15

q

VaR (q)

0.0 0.2 0.4 0.6 0.8 1.0

−0.10 −0.05 0.00 0.05 0.10 0.15

Litecoin

Euro

0.0 0.2 0.4 0.6 0.8 1.0

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15

q

VaR (q)

0.0 0.2 0.4 0.6 0.8 1.0

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15

MaidSafeCoin

Euro

0.0 0.2 0.4 0.6 0.8 1.0

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15

q

VaR (q)

0.0 0.2 0.4 0.6 0.8 1.0

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15

Monero

Euro

0.0 0.2 0.4 0.6 0.8 1.0

−0.15 −0.10 −0.05 0.00 0.05 0.10

q

VaR (q)

0.0 0.2 0.4 0.6 0.8 1.0

−0.15 −0.10 −0.05 0.00 0.05 0.10

Ripple

Euro

Figure 4.

Value at risk for the best ﬁtting distributions for daily log returns of the exchange rates of

Bitcoin (ﬁrst row, left), Dash (ﬁrst row, right), Dogecoin (second row, left), Litecoin (second row, right),

MaidSafeCoin (third row, left), Monero (third row, right) and Ripple (last row) from 23 June 2014 until

28 February 2017. Also shown are the values at risk for daily log returns of the exchange rates of the

Euro computed using the same best ﬁtting distributions.

J. Risk Financial Manag. 2017,10, 12 18 of 23

0.0 0.2 0.4 0.6 0.8 1.0

−0.08 −0.06 −0.04 −0.02 0.00

q

ES (q)

0.0 0.2 0.4 0.6 0.8 1.0

−0.08 −0.06 −0.04 −0.02 0.00

Bitcoin

Euro

0.0 0.2 0.4 0.6 0.8 1.0

−0.15 −0.10 −0.05 0.00

q

ES (q)

0.0 0.2 0.4 0.6 0.8 1.0

−0.15 −0.10 −0.05 0.00

Dash

Euro

0.0 0.2 0.4 0.6 0.8 1.0

−0.12 −0.10 −0.08 −0.06 −0.04 −0.02 0.00

q

ES (q)

0.0 0.2 0.4 0.6 0.8 1.0

−0.12 −0.10 −0.08 −0.06 −0.04 −0.02 0.00

Dogecoin

Euro

0.0 0.2 0.4 0.6 0.8 1.0

−0.12 −0.10 −0.08 −0.06 −0.04 −0.02 0.00

q

ES (q)

0.0 0.2 0.4 0.6 0.8 1.0

−0.12 −0.10 −0.08 −0.06 −0.04 −0.02 0.00

Litecoin

Euro

0.0 0.2 0.4 0.6 0.8 1.0

−0.15 −0.10 −0.05 0.00

q

ES (q)

0.0 0.2 0.4 0.6 0.8 1.0

−0.15 −0.10 −0.05 0.00

MaidSafeCoin

Euro

0.0 0.2 0.4 0.6 0.8 1.0

−0.20 −0.15 −0.10 −0.05 0.00

q

ES (q)

0.0 0.2 0.4 0.6 0.8 1.0

−0.20 −0.15 −0.10 −0.05 0.00

Monero

Euro

0.0 0.2 0.4 0.6 0.8 1.0

−0.15 −0.10 −0.05 0.00

q

ES (q)

0.0 0.2 0.4 0.6 0.8 1.0

−0.15 −0.10 −0.05 0.00

Ripple

Euro

Figure 5.

Expected shortfall for the best ﬁtting distributions for daily log returns of the exchange rates

of Bitcoin (ﬁrst row, left), Dash (ﬁrst row, right), Dogecoin (second row, left), Litecoin (second row,

right), MaidSafeCoin (third row, left), Monero (third row, right) and Ripple (last row) from 23 June 2014

until 28 February 2017. Also shown are the expected shortfall values for daily log returns of the

exchange rates of the Euro computed using the same best ﬁtting distributions.

J. Risk Financial Manag. 2017,10, 12 19 of 23

4.6. Kupiec’s test

The

p

-values of Kupiec’s test for the best ﬁtting distribution for each of the seven cryptocurrencies

are shown in Figure 6. The

p

-values (points above 0.05 in Figure 6) suggest that the out of sample

performance of VaR can be considered accurate at the corresponding values of

q

for any of the best

ﬁtting distributions. The motivation for using the Kupiec’s test is that we can give predictions of

the exchange rate of the different cryptocurrencies, including predictions for the extreme worst case

scenario and the extreme best case scenario.

0.90 0.92 0.94 0.96 0.98 1.00

0.0 0.2 0.4 0.6 0.8 1.0

q

p−value

0.90 0.92 0.94 0.96 0.98 1.00

0.2 0.4 0.6 0.8 1.0

q

p−value

0.90 0.92 0.94 0.96 0.98 1.00

0.0 0.2 0.4 0.6 0.8 1.0

q

p−value

0.90 0.92 0.94 0.96 0.98 1.00

0.4 0.6 0.8 1.0

q

p−value

0.90 0.92 0.94 0.96 0.98 1.00

0.2 0.4 0.6 0.8 1.0

q

p−value

0.90 0.92 0.94 0.96 0.98 1.00

0.2 0.4 0.6 0.8 1.0

q

p−value

Figure 6. Cont.

J. Risk Financial Manag. 2017,10, 12 20 of 23

0.90 0.92 0.94 0.96 0.98 1.00

0.2 0.4 0.6 0.8 1.0

q

p−value

Figure 6.

Kupiec’s

p

-values for the best ﬁtting distributions for daily log returns of the exchange rates

of Bitcoin (ﬁrst row, left), Dash (ﬁrst row, right), Dogecoin (second row, left), Litecoin (second row,

right), MaidSafeCoin (third row, left), Monero (third row, right) and Ripple (last row) from 23 June 2014

until 28 February 2017.

4.7. Dynamic Volatility

So far in this section, we have supposed that volatility is represented by a ﬁxed parameter of

a distribution. Often in ﬁnancial series, volatility varies throughout time. This could be accommodated

in various ways. One is to treat volatility itself as a random variable. Another is to let volatility depend

on some covariates including time as in GARCH models for example. In the ﬁrst case, the interest will

be on the distribution of volatility.

For the seven cryptocurrencies and the Euro, we computed standard deviations of daily log

returns of the exchange rates over windows of width 20 days. Histograms of these standard deviations

are shown in Figure 7. We see that the distribution is skewed for the seven cryptocurrencies and the

Euro. The range of volatility appears smallest for the Euro and second smallest for Bitcoin. The range

appears largest for Dogecoin, Litecoin and Monero.

Bitcoin

SD

Frequency

0.01 0.03 0.05 0.07

0 6

Dash

SD

Frequency

0.02 0.06 0.10 0.14

0 6

Dogecoin

SD

Frequency

0.00 0.05 0.10 0.15

0 6 14

Litecoin

SD

Frequency

0.00 0.05 0.10 0.15

0 6

MaidSafecoin

SD

Frequency

0.02 0.06 0.10

0 6

Monero

SD

Frequency

0.05 0.10 0.15

0 6

Ripples

SD

Frequency

0.00 0.04 0.08 0.12

0 10

Euro

SD

Frequency

0.000 0.005 0.010 0.015

0 40

Figure 7.

Histograms of standard deviations of daily log returns of the exchange rates of the seven

cryptocurrencies over windows of width 20 days. Also shown is the histogram of standard deviations

of daily log returns of the exchange rates of the Euro over windows of width 20 days.

J. Risk Financial Manag. 2017,10, 12 21 of 23

5. Conclusions

We have analyzed the exchange rate of the top seven cryptocurrencies versus the U.S. Dollar using

eight of the most popular parametric distributions in ﬁnance. From our analysis of over two and a half

years of data, it is clear that most cryptocurrencies exhibit heavy tails. Using the discrimination criteria

of the log likelihood, AIC, AICc, BIC, HQC and CAIC, the results obtained show that none of the

distributions used give the best ﬁt jointly across the data for all of the cryptocurrencies. Instead, we ﬁnd

that the generalized hyperbolic distribution gives the best ﬁt for the Bitcoin and LiteCoin; the normal

inverse Gaussian distribution gives the best ﬁt for Dash, Monero, and Ripple; the generalized

t

distribution gives the best ﬁt to Dogecoin; the Laplace distribution gives the best ﬁt to MaidSafeCoin.

Implications of these results are in the area of risk management, where one may need to compute

the VaR and ES for risk, but also for investment purposes. To our knowledge, this is the ﬁrst study

investigating the statistical properties of cryptocurrencies, going beyond Bitcoin and the traditional

currencies. Indeed, there is much scope for future work, and possible extensions could include: (i) using

GARCH type processes to model the log returns of cryptocurrencies, for example the distributions

mentioned in Section 3can be used for modeling the innovation processes; (ii) using multivariate

processes to model the joint distribution of the log returns of cryptocurrencies; (iii) using nonparametric

or semiparametric distributions to analyze the exchange rates of cryptocurrencies.

Acknowledgments:

Joerg Osterrieder gratefully acknowledges support from the Swiss State Secretariat for

Education, Research and Innovation within the context of the European COST network “Mathematics for Industry”.

All authors would like to thank the two referees and the Editor for careful reading and comments which greatly

improved the paper.

Author Contributions: All authors contributed equally to the work.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

Akaike, Hirotugu. 1974. A new look at the statistical model identiﬁcation. IEEE Transactions on Automatic Control

19: 716–23.

Anderson, D. R., K. P. Burnham, and G. C. White. 2010. Comparison of Akaike information criterion and consistent

Akaike information criterion for model selection and statistical inference from capture-recapture studies.

Journal of Applied Statistics 25: 263–82.

Anderson, Theodore W., and Donald A. Darling. 1954. A test of goodness of ﬁt. Journal of the American Statistical

Association 49: 765–69.

Aquino, Rodolfo Q. 2006. Efﬁciency of the Philippine stock market. Applied Economics Letters 13: 463–70.

Azzalini, Adelchi, and Antonella Capitanio. 2003. Distributions generated by perturbation of symmetry with

emphasis on a multivariate skew tdistribution. Journal of the Royal Statistical Society B 65: 367–89.

Barndorff-Nielsen, Ole. 1977. Exponentially decreasing distributions for the logarithm of particle size. Proceedings

of the Royal Society of London Series A Mathematical and Physical Sciences 353: 401–9.

Bitcoin Project. 2017. Frequently Asked Questions. Available online: https://bitcoin.org/en/faq#what-is-bitcoin

(accessed on 2 February 2017).

Bozdogan, Hamparsum. 1987. Model selection and Akaike’s Information Criterion (AIC): The general theory and

its analytical extensions. Psychometrika 52: 345–70.

Breusch, Trevor S., and Adrian R. Pagan. 1979. A simple test for heteroscedasticity and random coefﬁcient

variation. Econometrica 47: 1287–94.

Briere, Marie, Kim Oosterlinck, and Ariane Szafarz. 2015. Virtual currency, tangible return: Portfolio diversiﬁcation

with Bitcoins. Journal of Asset Management 16: 365–73.

Bruneau, Gabriel, and Kevin Moran. 2017. Exchange rate ﬂuctuations and labour market adjustments in Canadian

manufacturing industries. Canadian Journal of Economics 50: 72–93.

Burnham, Kenneth P., and David R. Anderson. 2004. Multimodel inference: Understanding AIC and BIC in model

selection. Sociological Methods and Research 33: 261–304.

J. Risk Financial Manag. 2017,10, 12 22 of 23

Chu, Jeffrey, Saralees Nadarajah, and Stephen Chan. 2015. Statistical analysis of the exchange rate of Bitcoin.

PLoS ONE 10: e0133678. doi:10.1371/journal.pone.0133678.

CoinMarketCap. 2017. Crypto-Currency Market Capitalizations. Available online: https://coinmarketcap.com/

(accessed on 2 February 2017).

Corlu, Canan G., and Alper Corlu. 2015. Modelling exchange rate returns: Which ﬂexible distribution to use?

Quantitative Finance 15: 1851–64.

Cramer, Harald. 1928. On the composition of elementary errors. Scandinavian Actuarial Journal 1928: 13–74.

Dai, Pham V., Sarath Delpachitra, and Simon Cottrell. 2017. Real exchange rate and economic growth in east Asian

countries: The role of ﬁnancial integration. Singapore Economic Review 62. doi: 10.1142/S0217590816500168

DogeCoin. 2017. DogeCoin. Available online: http://dogecoin.com/ (accessed on 2 February 2017).

Durbin, James, and Geoffrey S. Watson. 1950. Testing for serial correlation in least squares regression I. Biometrika

37: 409–28.

Durbin, James, and Geoffrey S. Watson. 1951. Testing for serial correlation in least squares regression II. Biometrika

38: 159–78.

Durbin, James, and Geoffrey S. Watson. 1971. Testing for serial correlation in least squares regression III. Biometrika

58: 1–19.

Fang, Yixin. 2011. Asymptotic equivalence between cross-validations and Akaike Information Criteria in

mixed-effects models. Journal of Data Science 9: 15–21.

Gosset, William Sealy. 1908. The probable error of a mean. Biometrika 6: 1–25.

Hannan, Edward J., and Barry G. Quinn. 1979. The determination of the order of an autoregression. Journal of the

Royal Statistical Society B 41: 190–95.

Hencic, Andrew, and Christian Gourieroux. 2014. Noncausal autoregressive model in application to Bitcoin/USD

exchange rate. In Econometrics of Risk. Berlin: Springer, pp. 17–40.

Hu, Shuhua. 2007. Akaike Information Criterion. Available online: http://citeseerx.ist.psu.edu/viewdoc/

download?doi=10.1.1.353.4237&rep=rep1&type=pdf (accessed on 2 February 2017).

Hurvich, Clifford M., and Chih-Ling Tsai. 1989. Regression and time series model selection in small samples.

Biometrika 76: 297–307.

Kinateder, Harald. 2015. What drives tail risk in aggregate European equity markets? Journal of Risk Finance

16: 395–406.

Kinateder, H. (2016). Basel II versus III—A comparative assessment of minimum capital requirements for internal

model approaches. Journal of Risk 18: 25–45.

Kolmogorov, A. 1933. Sulla determinazione empirica di una legge di distribuzione. Giornale dell’Istituto Italiano

degli Attuari 4: 83–91.

Kristoufek, Ladislav. (2015). What are the main drivers of the Bitcoin price? Evidence from wavelet coherence

analysis. PLoS ONE 10: e0123923. doi: 10.1371/journal.pone.0123923

Laplace, P. -S. 1774. Mémoire sur la probabilité des causes par les évènements. Mémoires de l’Academie Royale des

Sciences Presentés par Divers Savan 6: 621–56.

Linden, Mikael. 2001. A model for stock return distribution. International Journal of Finance and Economics 6: 159–69.

Linden, Mikael. 2005. Estimating the distribution of volatility of realized stock returns and exchange rate changes.

Physica A—Statistical Mechanics and Its Applications 352: 573–83.

Litecoin Project. 2017. Litecoin—Litecoin Wiki. Available online: https://litecoin.info/Litecoin (accessed on

2 February 2017).

MaidSafe. 2017a. MaidSafe—The New Decentralized Internet. Available online: https://maidsafe.net/ (accessed

on 2 February 2017).

MaidSafe. 2017b. MaidSafe—SafeCoin. Available online: https://maidsafe.net/safecoin.html (accessed on

2 February 2017).

McDonald, James B., and Whitney K. Newey. 1988. Partially adaptive estimation of regression models via the

generalized tdistribution. Econometric Theory 4: 428–57.

McNelis, Paul D. 2005. Neural Networks in Finance: Gaining Predictive Edge in the Market. Cambridge: Elsevier.

Monero. 2017. About Monero. Available online: https://getmonero.org/knowledge-base/about (accessed on

2 February 2017).

Nadarajah, Saralees, Emmanuel Afuecheta, and Stephen Chan. 2015. A note on “Modelling exchange rate returns:

Which ﬂexible distribution to use?” Quantitative Finance 15: 1777–85.

J. Risk Financial Manag. 2017,10, 12 23 of 23

Natenberg, Sheldon. 2007. Option Volatility Trading Strategies. Hoboken: John Wiley & Sons.

Parlapiano, Fabio, Vitali Alexeev, and Mardi Dungey. 2017. Exchange rate risk exposure and the value of European

ﬁrms. European Journal of Finance 23: 111–29.

Podobnik, Boris, Davor Horvatic, Alexander M. Petersen, and H. Eugene Stanley. 2009. Quantitative relations

between risk, return and ﬁrm size. EPL 85: 50003.

R Development Core Team. 2017. R: A Language and Environment for Statistical Computing. Vienna: R Foundation

for Statistical Computing.

Ripple. 2017. Welcome to Ripple. Available online: https://ripple.com/ (accessed on 2 February 2017).

Svetlana Sapuric, and Angelika Kokkinaki. 2014. Bitcoin is volatile! Isn’t that right? In Business Information Systems

Workshops, Lecture Notes in Business Information Processing. Berlin: Springer, pp. 255–65.

Schroeder, Marcel. 2017. The equilibrium real exchange rate and macroeconomic performance in developing

countries. Applied Economics Letters 24: 506–9.

Schwarz, Gideon. 1978. Estimating the dimension of a model. Annals of Statistics 6: 461–64.

Seyyedi, Seyyedsajjad. 2017. Analysis of the interactive linkages between gold prices, oil prices, and exchange

rate in India. Global Economic Review 46: 65–79.

Smirnov, Nickolay. 1948. Table for estimating the goodness of ﬁt of empirical distributions. Annals of Mathematical

Statistics 19: 279–81.

The Dash Network. 2017. What is Dash?—Ofﬁcial Documentation. Available online: https://dashpay.atlassian.

net/wiki/pages/viewpage.action?pageId=1146914 (accessed on 2 February 2017).

Von Mises, Richard E. 1928. Wahrscheinlichkeit, Statistik und Wahrheit. Heidelberg: Julius Springer.

Vrieze, Scott I. 2012. Model selection and psychological theory: A discussion of the differences between the Akaike

information criterion (AIC) and the Bayesian information criterion (BIC). Psychological Methods 17: 228–43.

Zhu, Dongming, and John W. Galbraith. 2010. A generalized asymmetric Student-

t

distribution with application

to ﬁnancial econometrics. Journal of Econometrics 157: 297–305.

c

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