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2017, Vol.

value behavtatistical pro

nagement, E

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4, No. 1

viour of operties.

Extreme


1. Intro

Since cryptocterms o(CoinMRipple, with thDash, A2016).

A crypcryptogcurrenccurrenc

RegulatWe wilinvestm

The main the fo

Table 1

The marates frhowevethe statcharacteinto cr

oduction an

Bitcoin bcurrencies hof total mar

MarketCap, representine top ten o

Augur, Maid

tocurrency graphy to secy. Cryptocucies.

tors need toll show th

ment or curre

arket capitalollowing Ta

. Market ca

Name

Bitcoin

Ethereu

Ripple

Litecoi

Moner

Ethereu

Dash

Augur

Nem

MaidS

ain purposeom a statis

er the actualtistical properistics. Th

ryptocurrenc

nd Motivat

became theave been crrket value, 2016). The

ng 7.6% andof them (BidSafeCoin,

is a digitaecure the traurrencies ar

o ask themsehat the riskency that is

lization of thable 1 (see C

apitalization

n

um

in

ro

um Classic

afeCoin

e of the papstical point l behavior operties of ce analysis ocies and y

ion

e first dereated. Bitcorepresenting

e second and 2.4% of thtcoin, EtheWaves) rep

al asset deansactions are a subset

elves if crypks of cryptavailable to

he top ten cCoinMarket

n of cryptocu

Market c

11350

932

293

185

89

80

67

53

37

36

per is to unof view. Th

of returns is cryptocurrenof extreme you want t

58

ecentralizedoin, as of Ng over 81%nd third larhe market. T

ereum, Ripppresenting ab

esigned to and to contt of alternat

ptocurrencitocurrencieo retail clien

cryptocurrentCap, 2016)

urrencies

cap (bn USD)

nderstand anhe underlyinot yet wel

ncies and wevents is pato understa

d cryptocuNovember 20% of the totrgest cryptoThere are mple, Litecoinbout 95% o

work as atrol the creative currenc

ies can freels are largents.

ncies as of N):

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nd characteing technolll understoowe focus inarticularly rand what r

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more than 72n, Ethereum

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ly be used ber than any

November 8

ket cap, relativ

%

%

%

%

%

%

%

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%

erize cryptoogical issue

od. We are thn particularrelevant if yrisks you

Finance andISSN 2

2017, Vol.

2009, nulargest of itsof cryptocuare Ethere

20 cryptocum Classic, Met (CoinMar

of exchangditional unitecifically o

by retail cuy other tra

8, 2016 can

ve

ocurrency ees are welltherefore der on extremyou are an are facing

d Banking 2374-2089

4, No. 1

umerous s kind in urrencies eum and urrencies Monero, rketCap,

ge using ts of the f digital

stomers. aditional

n be seen

xchange -known,

escribing me value

investor . For a


characteOsterriethem toexchang

The papused ancryptocand resextremethat wedifferenindex. Fcomputvalue difindings

1.1 Wha

A cryptand maCryptocmoney/bitcoin’

1.2 A Sh

In our sare: Bitbecausethe retu

Bitcoin all funcout colgovernmensure digitallyalgorithfrom a designerather th

LitecoinprocessInternetefficien

erization ofeder & Loro the G10 cge rate retur

per is organnd the sourccurrencies, cults from ee events wite own crypnt currencieFurthermorted. A closeistribution ws.

at are Cryp

tocurrency ianaged throcurrencies /centralized ’s blockchai

hort Descrip

study, we atcoin, Litece they have urns.

(BTC) is actions such llectively bment manipthat things

y through ahms and crufiat currenc

ed around thhan relying

n (LTC) issing smallert currency

ntly mined

f the paramerenz (2016)currencies. Frns and con

nized as folces from whcomputes vextreme valth a particu

ptocurrenciees and comre, two imper look at thwill be mad

tocurrencie

is a digital aough the us

use decebanking s

in transactio

ption of the

analyze six coin, Ripplexisted on

a decentralias currency

by the netwpulation or s run smooa “mining”unch numbecy, which is he idea of uon central a

s regarded r transactionbased on th

with cons

etric distribu, the authorFor the ana

nsider the qu

llows. In sehich it was volatilities aue theory t

ular focus ones. We use

mpute the taportant risk he generaliz

de in section

es?

asset designe of advan

entralized systems. Thon database

e Individual

out of the tle, Monero,or before J

ized currency issuance, work. Whilinterference

othly or to ” process thers. These cbacked by

sing cryptoauthorities.

as Bitcoin’ns faster. Ithe Bitcoin sumer-grad

59

utions of crrs analyze

alysis fiat cuestion, wha

ection 2, weretrieved. S

and correlatto analyze tn the left tacopula the

ail dependek measures, zed Pareto n 5. The las

ned to worknced encryp

control ashe decentr

e in the role

Cryptocurr

ten largest c, MaidSafeJune 2014, s

cy that usetransaction le this dece, the flipsiback the v

hat requirescharacteristthe full faitgraphy to c

’s leading t was foundprotocol bu

de hardware

ryptocurrencthe Bitcoin

currencies, Nat flexible d

e give an ovSection 3 lotions. In Sethe behavioail of the diseory to looknce coefficvalue-at-ridistributiont section 6 c

k as a mediuption technis opposedalized contof a distribu

rencies

cryptocurreeCoin, Dashso that we h

s peer-to-peprocessing

centralizatioide is that tvalue of a s powerful ics make Bth and credi

control the c

rival at prded in Octout differs fe. Litecoin

International

cies, see Osn exchange Nadarajah edistribution t

verview of ooks at statiection 4 weour of cryptstribution, ik at the de

cients as wesk and expn and the gconcludes a

um of exchaiques knowd to centtrol is relauted ledger.

ncies. The h. We havehave suffici

eer technolog and verificon renders here is no cBitcoin. Bcomputers

Bitcoin fundit of its govcreation and

esent, and ober 2011. from Bitcoinn provides

Finance andISSN 2

2017, Vol.

sterrieder (2rates and c

et al. (2015to use.

f the data wistical prope will use ctocurrenciesimplicitly aependence bell as the epected shorgeneralized and summar

ange that iswn as cryptotralized elated to the.

selected cue chosen thient data to

ogy, which cation to beBitcoin frecentral auth

Bitcoins are s to solve cdamentally dvernment. Bd transfer of

it is desigIt is a peer

in in that itfaster tran

d Banking 2374-2089

4, No. 1

2016). In compare

5) model

which we erties of concepts s during ssuming between extremal rtfall are extreme

rizes our

s created ography. lectronic

use of

urrencies hose six analyze

enables e carried ee from hority to

created complex different

Bitcoin is f money,

gned for r-to-peer t can be nsaction


confirmtarget thwas to used to

Dash (cryptocDash odecentrfor theAlgorith

MoneroprivacyBitcoindifferen

Ripple Chris Lcurrencof the mutilizing

MaidSaNetworan algorcoin wibeing exSafecoiresourcconnectplatform

2. Data

We are from quprovidindifferencryptocdownlochosen cryptoc

The topAugur, cryptoc

mations andhe regular cprovide a mmine bitcoi

(formerly currency thaoperates a alized auto proof-of-whm family)

o (XMR) isy, decentrali, Monero is

nces relating

(XRP) wasLarsen in 20cy componemain contesg collective

afeCoin (Sark. Safecoinrithm. Onlyill have its oxchanged foins are givees to the ntivity, this pm is referred

a

using histouandl.com, ng a weight

nt online excurrencies boaded from

to start onlcurrencies by

p ten of themMaidSafeC

currencies t

d uses a mcomputers amining algoins.

known as at offers indecentraliz

nomous orgwork. Insteor scrypt it

s an open ssation and ss based on g to blockch

s launched b012. Like Bnt is XRP, wstants for th

ely-trusted s

afecoin) is ans are beingy 4.3 billionown uniquefor network en to users etwork. Thprocess of pd to as farm

orical globawhich showted average

xchanges, cobased on act

June 23, 2ly in June y market ca

m (Bitcoin,Coin, NEM)that we hav

emory-hardand GPUs morithm that

Darkcoinnstant transzed governganization. ead of usint uses 11 rou

ource cryptscalability. Uthe Crypto

hain obfusca

by OpenCoBitcoin, Ripwhich has ahe title of Bubnetworks

a digital cryg distributedn MaidSafeCe identity. Fuservices, soin return foese includeproviding y

ming.

al price indws aggrega

e cryptocurrollate it andtivity, tradin2014 until t2014, becau

apitalization

, Ethereum,) represent mve chosen

60

d, scrypt-bamost people

could run

n and XCsactions, prnance and

Dash uses ng the SHunds of diff

tocurrency Unlike man

oNote protoation.

oin, a comppple is botha mathematiBitcoin sucs within the

yptocurrencd entirely aCoins will eurthermore o there will for them proe free storagyour resourc

dices for crated cryptocrency price.d calculate ang volume,the end of use that all

n as of Nove

, Ripple, Limore than 9out of the

ased miningalready havat the same

oin) is anivate transabudgeting a chained A-256 (fro

ferent hashin

created in ny cryptocucol and pos

pany foundeh a currencyical foundatccessor. It u larger netw

cy token thautomated w

ever be in ciSafecoins wbe a new suoviding somge space, pces and earn

ryptocurrenccurrency pr. They regua weighted liquidity aSeptember

lows us to ember 2016

itecoin, Ethe95% of the e top ten a

International

g proof-of-ve. One of the time, on t

n open soactions andsystem, m

hashing algom well-knng functions

April 2014 rrencies thassesses sign

ed by techny and a paytion like Bituses a consework.

at is in the with no humirculation atwill be recyupply of coime of their processing pning Safeco

cies from thrices from mularly collecaverage prind other far 2016. On analyze six

6.

ereum Clasmarket cap

are: Bitcoin

Finance andISSN 2

2017, Vol.

-work algorthe aims of the same h

ource peerd token funmaking it tgorithm calnown Securs.

that is focat are derivanificant algo

nology entreyment systetcoin. Ripplensus algor

heart of thman intervent one time a

ycled when ins for usersunused com

power, and oin in return

the databasemultiple exct informatiice for the d

actors. Dailyn purpose, wx out of the

ssic, Moneropitalization. n, Dash, L

d Banking 2374-2089

4, No. 1

rithm to Litecoin

hardware

r-to-peer ngibility. the first led X11 re Hash

cused on atives of orithmic

epreneur em. The le is one rithm by

he SAFE ntion by and each they are s to earn. mputing Internet

n on the

e BNC2 xchanges ion from different y data is we have e top ten

o, Dash, The six

LiteCoin,


MaidSa30, 201only cathe mar

3. Stati

Before properti

3.1 Dist

We wanDeviatichoice histograshows Bitcoin/with ba

Figu

The nexthe standistribu

afeCoin, Mo5, Ethereum

ame into exirket capitali

istical Prop

we go into ies of crypto

tribution of

nt to showons of the of using exam of daily a normal d/ USD exch

andwidth mu

ure 1. Histo

xt Figure 2 ndard norm

ution both on

onero and Rm Classic, wistence in 2zation as of

perties of C

the extremeocurrencies

f Returns

w the distribcryptocurre

xtreme valureturns of t

distribution hange ratesultiple 0.5. W

ogram of Bi

shows the mal distribun the left an

Ripple. We hwhich only 2015. In totaf November

ryptocurre

e value behs based on th

bution of reency distrib

ue theory tothe Bitcoin

with mean. The blue We see a su

tcoin/USD Gauss

qq-plot of ution. Againd the right

61

have omittestarted tradal, our choir 2016.

encies

haviour of crheir returns

eturns and bution fromo describe th

/ USD exchn and standline is show

ubstantial de

exchange rsian KDE ov

the empiricin, we obstail.

ed Ethereumding in 2016ice of crypt

ryptocurren.

compare thm the normahe tails. In hange rate. dard deviatwing a Gaueviation from

ate with fittverlaid

cal Bitcoin rserve large

International

m, with an in6, Augur andocurrencies

ncies, we an

hem to a nal distributiFigure 1, wThe red lineion taken f

ussian kernem the norm

ted normal d

returns versdeviations

Finance andISSN 2

2017, Vol.

nitial released NEM, whs represents

nalyze the st

normal distrion will juswe are showe which is ofrom the emel-density e

mal distributi

distribution

sus the quans from the

d Banking 2374-2089

4, No. 1

e in July hich also 88% of

tatistical

ribution. stify our wing the overlaid, mpirical stimator ion.

n and

ntiles of normal


,0.2 -t1

CWA_BTC: histogram ot daity r•lums

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Fi

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Figure

Figure

igure 2. QQ

llowing fivcurrencies, Dthe x-axis t

3. Histogram

4. Histogra

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ve Figures 3Dash, Litecto returns be

m of Dash/U

am of LTC/U

e Bitcoin/US

3, 4, 5, 6, coin, MaidSetween -0.2

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ange rate witKDEoverlai

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th fitted norid

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International

us the norm

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g occasional

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2017, Vol.

mal distributi

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bution and G

ution and G

d Banking 2374-2089

4, No. 1

ion

ve other hosen to ers.

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00 0' 02

Fig

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Figure

We see five chaof the distribu

gure 5. Histo

gure 6. Hist

7. Histogra

a substantiarts, 8, 9, 10standard no

ution both on

ogram of M

togram of X

m of XRP/U

ial deviation0, 11, 12 wormal distrin the left an

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ibution. Agnd the right

63

exchange rasian KDEov

exchange rasian KDEov

nge rate witKDEoverlai

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ate with fitteverlaid

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Finance andISSN 2

2017, Vol.

distribution

distribution a

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ts. In the foversus the q

ns from the

d Banking 2374-2089

4, No. 1

and

and

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00 01 02

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~ilyrnffls

Fi

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Fig

igure 8. QQ

Figure 9. QQ

gure 10. QQ

Q-Plot of the

Q-Plot of th

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e DASH/US

he LTC/USD

e MAID/US

64

SD exchang

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SD exchang

ge rate versu

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ge rate versu

International

us the norma

s the normal

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Finance andISSN 2

2017, Vol.

al distributi

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mal distribut

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4, No. 1

on

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3.2 Vola

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Table 2

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igure 11. QQ

igure 12. Q

atility

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owing Table

. Annualize

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Q-Plot of th

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here the volThe annualize 2:

ed volatility

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Bitcoi

Dash/U

Liteco

Maid/U

Moner

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igure 13, wersus the U

he XMR/US

he XRP/US

latilities of ozed standar

ange rate

n/USD

USD

oin/USD

USD

ro/USD

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we have plotSD.

65

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our cryptocrd deviation

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62%

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tted the 90-

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urrencies frn of returns

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day rolling

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us the norma

s the norma

rom June 20over this pe

volatility (

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2017, Vol.

al distributi

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(annualized)

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4, No. 1

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Table 3

Table 4

serve that crthe establish

rrelations

eresting quies look likn’s correlatition coefficr instance (tion of the 06 to 2016/0

. Kendall co

Bitco

DashLitec

MAI

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. Spearman

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Figure 13.

ryptocurrenhed fiat curr

uestion regke, and we on coefficie

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six crypto09).

orrelation fo

Bitco

oin 1

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n correlation

Bitcooin 1

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coin 0.753ID 0.278

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ple 0.183

90-days rol

ncies have erencies or a

arding crypstart by st

ent , Speaefinitions an10) and (Shocurrencies

or cryptocur

oin Dash

0.255

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n for cryptoc

oin Dash0.362

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66

lling volatil

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rrencies

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lity (annuali

olatility whtandard fina

ies is howyptocurrenck correlationisons of the07). Tables

USD) in our

Maid M

0.192 0.

0.142 0.0.126 0.

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ized, in pct)

ich is substancial assets

their muly correlation coefficienese three cos 3, 4 and r study ove

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d 5 summaer our full

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4, No. 1

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arize the dataset


2015

90-ditys rolling volo1tllity (o1nnui1Uzed, In pct)

2016

Date

Table 5

Overallshowing

Here, thstatistic

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(Gandacrypto-cprior towinner-substitupeople positiveif overavarious provide

. Pearson co

Bitco

Dash

Litec

MAI

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l, the levels g the highes

he technicalcal propertie

g at the timthe rolling 9

n, correlation

l & Halabcurrency spo ours): rei-take-all raution (crypseek altern

e correlationall sentimen

cryptocurre alternativ

orrelation fo

Bitco

oin 1

h 0.393

coin 0.611

ID 0.337

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ple 0.152

of correlatist correlatio

l similaritiees.

me-evolution90 day correns vary subs

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burda, 201pace to twoinforcemence against

ptocurrencienative invesn as cryptocnt for cryptrencies are es to Bitc

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0.393

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ewhat modeon coefficien

Litecoin ar

more compleC) to the oth

ng correlati

this time-beit investigific cryptoces, leading assets—popcurrency gegain in marground). T

milar in techx what the

International

Monero Rip

353 0.1

175 0.1

212 0.1

165 0.0

0.0

047 1

est—with Bnt of approx

re also clear

ex picture eher 5 crypto

on

changing cgating a difcurrencies o

to negativpularized byets too exprket value, ohis is due t

hnology andeir develop

Finance andISSN 2

2017, Vol.

pple

152

110

103

089

047

Bitcoin and ximately 0.6

rly reflected

emerges. Fiocurrencies

correlation fferent timeoutperformive correlatiy Bitcoin-, ipensive—leaor collectivto the fact d functiona

pers though

d Banking 2374-2089

4, No. 1

Litecoin 6).

d in their

igure 14 . As can

in the e period ing in a on) and in which ading to ely drop that the

ality, but ht to be


shortcomincentiv

In our somewhdroppineffects bfour: Lcryptocpointing

4. Extre

Apart frwe alsorisk mamultipleas the dextremebehavio

4.1 Cop

Copulasdependedecompspecificand, forfor all

A copudistribucontinuprobabiThen and in tthe reprtheoremfunction, … ,exponenbivariatallows tdistribu

mings of thve structure

study perihat stable—ng in the sebetween Bit

Litecoin seecurrencies ag towards th

eme Value

from lookino want to ananagement ae cryptocurrdependencee value theour under ta

pulas

s allows onence from posed into itc, let r let den, …

ula of thutions , …uous randomility integra, … ,this case a cresentation

m: given ann define. For exntially distrte distributithe research

utions, leavi

he original of mining n

od, we can—whereas th

econd half, tcoin and Lems to beare less corhese cryptoc

Analysis of

ng at the pronalyze extrand investmrencies, it is

e of returnseory to anail events.

ne to focuthe univarits univar, … ,note the ma, ∈ ,

e random v… , to them variablesal transform, ~ .copula is anin (1) parti

ny collectioed by (1) abxample, oneributed varion. The abher to direcng only the

Bitcoin prnew coins in

n see the Bhe correlatio

especially Litecoin, ande perceivedrrelated to currencies g

f Cryptocu

operties of treme eventsment purposs important in extremealyse the c

s on the diate distriburiate margin be a rand

arginal distr

vector is joint distris, the copu

ms of the or It is well

ny joint distrcularly usef

on of univarove definese might coriable via ability to coctly draw on task of mo

68

rotocol (forn the netwo

Bitcoin/Liteon of Bitcoin 2016. T

d reinforcemd as the m

Bitcoin—agiving way

urrencies

the cryptocs. Large losses. Furtherm

to considere scenarioscorrelation

dependence utions. An nal distributdom vector ribution of

,s thus a funibution , ula is uniqriginal vari

known tharibution funful for appl

ariate distribs a valid joinombine a

a Clayton combine marn the large deling the d

r instance, ork).

ecoin correoin to the oThis points ment in favomost viableand even nto the other

currency retsses or gainmore, whenr their corre. We will uof cryptoc

of random-dimensio

tions and anwith cumu. Then th

… ,nction that and we wr

que. It is ables, whicat if is nction with lied researchbutions ,nt distributinormally

copula, andrginal distribody of res

dependence

International

slow transa

elation beinother four c

to possiblyor of these twe substituteegatively cr two.

urns over thns are particn considerinlation, com

use techniqucurreny retu

m variables onal joint dn -dimensulative distrhere exists

maps the urite ~the joint d

ch are defincontinuous,0,1 mahers is the c… , and ion with madistributed

d obtain an ibutions witsearch on mstructure.

Finance andISSN 2

2017, Vol.

action time

ng rather hcrypto-currey mild subwo versus t

e, while thcorrelated a

he entire spcularly releng an invest

mmon movesues and tooturns as we

by separadistributionsional copulribution funa copula su

univariate m, … ,distributionned as , then ~argins. Whaconverse ofany copula

arginal distrvariable w

n unusual bith a copulamodeling un

d Banking 2374-2089

4, No. 1

s or the

high and encies is stitution

the other he other at times,

pectrum, evant for tment in s as well ols from ell their

ating the n can be la. To be nction uch that,

(1)

marginal . For n of the

. ~ 0,1 at makes f Sklar’s a , the ributions with an

but valid a model nivariate


Definitiif is cube

-dimen

• ,• 1,…• is

-volum

where t

For ins0, ,Copulasdownsidmultiva

In risk/pthat ardownsid

As far widely in the p

4.1.1 Em

We starcopula.

Definitiinvestigfrom a

copula

Howeveconstru

ion 1 (Copua joint cum0,1 witnsional cop, … , , 0,… ,1, , 1, … -non-deme of is

the

stance, in , 00 for

s are usefude regimes

ariate probab

portfolio mre especialde events m

as derivativused in app

pricing of co

mpirical Co

rt our copul

ion 2 (Emgate the un

random ve

observat

er, the marguct pseudo ∑

ulas). In promulative distth uniformpula if , , … ,… ,1 , th

ecreasing, inon-negati

# :the bivari0 , 1all 0

ul in portfoby allowinbility mode

management,ly importa

may occur (e

ves pricing plications oollateralized

opula

la analysis

mpirical copderlying coector (X1,X

tions wou

ginal distribcopula o

i

obabilistic ttribution fun

m margina

0, the

he copula is

i.e., for eave:

∈. iate case, 1, 1olio and risg the model separately

, copulas arant during e.g., the glob

is concernf financial

d debt obliga

by looking

pula). Whenopula. SuppX2,…,Xd) w

uld be (

bution funcobservations

instead. The

69

terms, : 0nction of a ls. In an

copula is ze

equal to u i

ach hyperre

,: 0,1, 1 a

and 0sk managemelling of they.

re used to p“downside

bal financia

ned, dependrisk assessmations (CDO

g at the sam

n studying pose we hawith continu

(U1i ,U2

i ,…,U

ctions ars by using

en, the pseu

0,1 → 0,-dimensio

nalytic term

ero if one o

if one argum

ectangle

10,1 → 0,

and ,1ment and he marginals

perform stree/crisis/panal crisis of 2

dence modement and acOs).

mple data an

multivariaave observauous margi

Udi )= (F1(X

re usually ng the emp

do copula o

International

1 is a -donal randomms, : 0,1f the argum

ment is u an∏ 0,

1 is a b1.

help us anaand depend

ess-tests andic regimes

2007-2008).

lling with cctuarial ana

nd construct

ate data, oations (X1

i ,Xins. The co

1i ),F2(X2

i ),…

not known. pirical dist

observations

Finance andISSN 2

2017, Vol.

dimensionam vector on1 → 0,1

ments is zero

nd all others, ⊆ 0,

bivariate co,alyse the efdence struct

d robustnesss” where .

copula funcalysis—for e

ting the un

one might X2

i ,…,Xdi ),i

orresponding

…,Fd(Xdi ) ,i

Therefore, tribution fu

s are define

d Banking 2374-2089

4, No. 1

l copula the unit

is a

o

s 1, 1 the

opula if ,ffects of ture of a

s checks extreme

ctions is example

derlying

want to =1,…,n, g "true"

=1,…,n.

one can functions

ed as


U1i,U2

i

then de

copula

Xki :Rk

i =

distribu

One of

Theoremfoundatcumulavector copula distribu

theoremis the Cunique : 0,1cumula

Visualizstructurempiricwe showshows mrelationexhibitsMaidSa

In the Litecoin

,…,Udi

= F

efined as C

samples c

=∑ nj=1 1 Xk

j ≤

ution of the

the most im

m 1 (Sklation for thetive distrib, , … ,

. Indeedution has a, … ,m also statesCartesian pr

if the mar→ 0,1 ative distribu

zing the emre of returnscal copula fow MaidSafemuch a highnship betwes very low afeCoin, Mo

following Fn.

F1n(X1

i ),F2n(X

Cn(u1,…,ud)

can also be

≤Xki . Ther

rank transfo

mportant the

ar). Sklar’s e applicatiobution func

can bed: …a density ⋅s that, giveroduct of thrginals

and marginution functio

mpirical copus. In the folor our six cr

feCoin, Monher correlat

een Bitcoin correlation

onero and R

Figure 15,

X2i ),…,Fd

n(X

)=1

n∑ n

i=1 1

e written a

refore, the

formed data

eorems in co

theorem non of copuction e expressed,, and this ⋅ … ⋅n , the co

he ranges ofare contins thon.

ula for crypllowing Figryptocurrennero and Riion than theand Litecoand large

Ripple.

we have p

70

Xdi ) ,i=1,…,n

U1i≤u1,…,U

as

empirical

a.

opula theory

named afteulas. Sklar’s, … ,d in terms of, … ,

is availab, where opula is uniof the margiuous. The

hen

ptocurrenciegures 15 anncies. First, ipple. We ce second on

oin and, to adispersion,

plotted the

n. The corr

Udi≤ud . Th/ , is

copula ca

y is Sklar’s

er Abe Sks theorem sℙf its margin

. Inble, it holds

is the ique on inal cdf’s. Tconverse i, … ,

es will help d 16 we wiwe have Bi

can see that ne. This is aa lesser ext

indicating

empirical c

International

responding

he compone

the rank o

an be seen

theorem.

klar, providstates that , … ,

nals n case thas further thdensity of ⋯

This impliesis also true

defines

us understaill show theitcoin, Dashthe first of

again an indtent, Dash.

low tail de

copula for

Finance andISSN 2

2017, Vol.

empirical c

ents of the

of the obs

n as the em

des the theevery mult

of a ℙat the multhat , …

f the copu⋯s that the ce: given as a -dime

and the depe three-dimh and Litecof those two dication of tThe secondependence b

Bitcoin, D

d Banking 2374-2089

4, No. 1

copula is

pseudo

ervation

mpirical

eoretical tivariate random and a

tivariate … ,ula. The

, which copula is a copula ensional

endence ensional oin, next copulas

the close d copula between

Dash and


In Figur

Overallcorrelat

4.1.2 G

We wansimultan

Definiti0,1 . probabicopula where the joinvector zanalyticand app

re 16, we ha

Figur

l, we can reptions than th

Gaussian Cop

nt to fit a paneously to t

ion 3 (GauIt is cons

ility integrawith param

is the nt cumulativzero and covcal formulaproximated

Figure 15.

ave plotted

re 16. Empir

produce ourhe other fou

pula

arametric cothe six cryp

ussian copultructed fro

al transformmeter matrix

inverse cumve distributvariance ma

a for the cousing nume

Empirical c

the empiric

rical copula

r earlier resur cryptocur

opula to theptocurrencie

la). The Gam a multiv

m. For a givx can bemulative distion functioatrix equal

opula functierical integr

71

copula for B

cal copula fo

a for MaidS

sult, that Bitrrencies.

e data. For tes under con

aussian copvariate norven correlate written as istribution fon of a multo the correion, ration. The

Bitcoin, Das

for MaidSafe

afeCoin, M

tcoin and Li

this, we usensideration.

pula is a drmal distribtion matrix

function of altivariate nelation matr

, it candensity can

International

sh, Litecoin

feCoin, Mon

onero and R

itecoin have

e the Gaussi

istribution bution over∈ 1,1a standard ormal distrrix . Whiln be upper n be written

Finance andISSN 2

2017, Vol.

n

nero and Rip

Ripple

e higher un

ian copula a

over the unr by us1 , the G, … ,normal and

ribution witle there is noor lower bas

d Banking 2374-2089

4, No. 1

pple.

derlying

and fit it

nit cube sing the

Gaussian ,

d is th mean o simple

bounded,


N 0

0 0

N 0

Q 0

00

"

.. CWU,TC

..

.. ..

Emplrtc.111 copul;a

. \ . Oo , O • II . (II

..

Empl ric.iil copul;a

.. "' "

where

In our Table 6used is

Table 6

The stan

Table 7

Bitc

Dash

Litec

MAI

Mon

Ripp

Also, foLitecoin

is the ide

case, this l6, together wmaximum l

. Coefficien

Bitco

Dash

Litec

MAI

Mon

Ripp

ndard errors

. Standard e

Bitc

oin 0

h 0.02

coin 0.0

ID 0.03

nero 0.02

ple 0.03

or visualizatn and Dash

1entity matrix

leads to a swith their stlikelihood a

nts of Gauss

Bitco

oin 1

h 0.391

coin 0.753

ID 0.315

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ple 0.185

s of our Gau

errors of Ga

coin D

0

28 0

12 0

30 0

28 0

33 0

tion purposin Figure 1

x.

six-dimensiotandard erro

and the log-

sian copula

oin Dash

0.391

1 1

3 0.326

5 0.219

0 0.213

5 0.160

usian copul

aussian copu

Dash L

0.028 0

0 0

0.029 0

0.032 0

0.032 0

0.034 0

es, we show7 as well as

72

12 ⋮onal copulaors in Tablelikelihood r

for cryptoc

Litecoin

0.753

0.326

1

0.220

0.280

0.179

la:

ula for cryp

Litecoin

0.012

0.029

0

0.032

0.030

0.033

w the three-s for MaidS

⋅a for whiche 7. The estresulted in 5

currencies

Maid M

0.315 0.

0.219 0.

0.220 0.

1 0.

0.201 1

0.120 0.

ptocurrencie

Maid

0.030

0.032

0.032

0

0.033

0.034

-dimensionaSafeCoin, Ri

International

⋅ ⋮h we recordtimation pro548.

Monero Rip

380 0.1

213 0.1

280 0.1

201 0.1

0.0

078 1

es

Monero

0.028

0.032

0.030

0.033

0

0.035

al Gaussian ipple and M

Finance andISSN 2

2017, Vol.

d the paramocedure wh

pple

185

160

179

120

078

Ripple

0.033

0.034

0.033

0.034

0.035

0

copula for Moneroin Fig

d Banking 2374-2089

4, No. 1

meters in hich was

e

Bitcoin, gure 18.


4.2 Tail

Apart frextremedependecoeffici(Jawors

The res

Figur

l Dependenc

from the core events isence of theients, usingski, Durante

ults for the

Figure 17.

re 18. Gauss

ce Coefficie

rrelation of s of utmose cryptocurrg the nonpae, & Hardle

lower tail d

Gaussian c

sian copula

ent

the cryptoct importanrency returnarametric e, 2010), com

dependence

73

copula for B

a for MaidSa

currency retnce. For thns. First, westimator omputed for a

can be seen

Bitcoin, Das

afeCoin, Mo

turns, lookinis purpose e compute f (Schmid all pairs of r

n in Table 8

International

sh, Litecoin

onero and R

ng at the bewe are co

the matrix & Schmid

returns.

:

Finance andISSN 2

2017, Vol.

Ripple

ehaviour of onsidering of tail-dep

dt, 2007) (s

d Banking 2374-2089

4, No. 1

f them in the tail endence see also


.. 0

" :; ,r_'

0 0

" 0

0 0

00

.. 0 0

0 Cl. CE: .. >< 0

,r_' ;: 0 Cl

:;i 0

00

0 0

0

0 0 <!>

02

0

00 0 0 0 0

0

0 0 Oo

0.2

o.•

GWA_BTC

0 • oo

• 0

o.•

GWA_t.WD

0

06

0

06

Gaussian copula

0 0 O 0

0.8

0

Gausslan copula

0

f0 0 0 o"

ooooo ooo 0~

• O 0 0 0 • • 0.8

'0

'0

0 0 0 0 oO 0

0 0

oO:t',l>OOOo.o 0 Oo

00 0~ O O O O 0 8 0 0

0 00 \Co O 4) oO 0 0 t" i: 00 oO i 0 0 oo

ooo 03'0 0 0 o 0 o O o o O O (9ct> o o

0 0 00 0 ß, 00 0 O 00 0 0 <f>O O q,> 0

0 ..1. 0 0 0 0 09 0 o o O<) 00 0 0 CE: • 4{,,.,~ qpo8 <P O 0 :ll • '0

>< s. 0 ,r_' • • ;:

<f Cl • 0

'0

Table 8

The cumin

Litecoinwith all

We are the datafound in

Table 9

Using afor Bitc

4.3 Valu

We wacryptocshortfal

Definitiportfoli

. Lower tail

Bitcoin

Dash

Litecoin

MAID

Monero

Ripple

ut-off paramn 100/nobsn have the l other crypt

also compua and returnn (Jaworski

. Lower tail

Bitcoin

Dash

Litecoin

MAID

Monero

Ripple

a t-copula, tcoin and Lit

ue-at-risk a

ant to characurrencies. Tll.

ion 4 (at Rio), then

l dependenc

Bitcoin

1

0.300

0.650

0.272

0.368

0.151

meter in s, 0.1 , werhighest tai

tocurrencies

uting the loning the impi, Durante, &

l dependenc

Bitcoin

1

0.140

0.479

0.102

0.053

0.046

there is onlytecoin.

nd Expected

acterize theThe two mo

Risk). Math is t

ce of crypto

Dash L

0.300 0

1 0

0.270 1

0.116 0

0.164 0

0.024 0

0,1 belore nobs stail dependens.

ower tail depplied tail-de& Hardle, 2

ce of crypto

Dash

0.140

1

0.082

0.048

0.100

0.001

y very low t

d Shortfall

e losses thaost importa

hematically,the negative

74

ocurrencies

Litecoin

0.650

0.270

1

0.187

0.318

0.156

ow which tands for thnce, wherea

pendence cependence c2010) and (S

ocurrencies,

Litecoin

0.479

0.082

1

0.033

0.020

0.048

tail depende

at can occuant tail risk

y, if is ae of the -q∈ |

Maid

0.272

0.116

0.187

1

0.177

0.079

the estimatihe number as Ripple ha

coefficient bcoefficient iSchmid & S

fitting a t-c

Maid

0.102

0.048

0.033

1

0.107

0.000

ency betwee

ur in extremmeasures a

a random vquantile, i.e

International

Monero

0.368

0.164

0.318

0.177

1

0.104

ion takes pof observaas the lowe

by fitting pain Table 9.

Schmidt, 200

copula

Monero

0.053

0.100

0.020

0.107

1

0.005

en all crypt

me events are value-at

variable (e.., 1 .

Finance andISSN 2

2017, Vol.

Ripple

0.151

0.024

0.156

0.079

0.104

1

place is chations. Bitcest tail dep

airwise t coDetails of i07).

Ripple

0.046

0.001

0.048

0.000

0.005

1

tocurrencies

when invet-risk and e

.g., the pri

d Banking 2374-2089

4, No. 1

hosen as coin and endence

opulas to it can be

s, except

esting in expected

ice of a

(2)


Definiti. F

of a loss

(Artznevalue-at

In Tablcryptocwhich cGaussiaobtaininexpecte

where

Table 1

We seealmost one daytranslatGaussiahigher tfrom th

In Tableusing twthe expe

Seconddistribuanalytic

ion 5 (shortFor a givens that excee

er, Delbaent-risk, is a c

le 10, we rcurrencies ucorrespondsan value-at-ng the meaed shortfall,

0.950. Value-at-

e that the vthree times

y, about once to losingan VaR, we than the his

he Gaussian

e 11 below, wo methodsected shortf

, the Gausution to thecally the exp

tfall). Expen random vaeds

n, Eber, & coherent risk

record the under consids to taking -risk, whichan and using the fo

5 denotes t

-risk for cry

Exchange

Bitcoin/U

Dash/USD

Litecoin/U

Maid/USD

Monero/U

Ripple/US

value-at-risk as large as

ce every 20 g more tha

can observtorical VaRdistribution

we record s: First, thefall from the

ssian expece data, obtapected shor

cted shortfaariable ,:

Heath, 19k measure.

value-at-risderation, usthe 95% q

h is computstandard d

formula

.the 95% q

yptocurrenci

e rate Hi

USD 0.0

D 0.

USD 0.0

D 0.

USD 0.

SD 0.

k for MaidSs the one fodays, whenn 4.8% on

ve that thoseR. This shown and the hig

the expectee historical e historical

cted shortfaaining the tfall, using

75

fall estimatethe expecte

|999) argue

sk at the 95sing two m

quantile of tted by first deviation

⋅quantile of t

ies, left tail

istorical VaR

048

106

068

125

125

101

SafeCoin anor Bitcoin. Yn you are invnce every 2e two numb

ws the deviagher risks th

ed shortfall expected shsample.

fall, which and stan

the formula

es the potened shortfall

that expec

5% level fomethods: Fir

the negativefitting a no, and then

0.95the standard

Gaussia

0.054

0.101

0.083

0.123

0.132

0.103

nd MoneroYou can expvested in M20 days. Cbers usuallyation of the hat they bea

at the 95% hortfall, wh

is computndard deviaa

International

ntial size of is defined a

cted shortfa

for Bitcoin st, the histoe exchangeormal distrin computin

d normal dis

an VaR

are the hipect to lose

Monero. For omparing h

y differ. Gaucryptocurrear.

level for ouhich corresp

ted by firstation , an

Finance andISSN 2

2017, Vol.

f the loss exas the expec

fall, as opp

and the otorical valuee rates. Secoibution to tng analytic

stribution.

ighest of ale more thanBitcoin, thihistorical Vussian VaR ency exchan

ur cryptocurponds to com

st fitting a nd then com

d Banking 2374-2089

4, No. 1

xceeding cted size

(3)

posed to

ther five e-at-risk, ond, the the data, ally the

ll, being n 12% in is would VaR and is much

nge rates

rrencies, mputing

normal mputing


where distribu

Table 1

We see Bitcoinlose motruly risGaussiaone wovalue-atthe riskrespecti

Table 1

Expecte

0.ution.

1. Expected

that the exp. Provided ore than 12%sky investman ES is lowould expect-risk and t

k of being loively.

2. Expected

ed shortfall

.95 denot

d shortfall fo

Exchan

Bitcoin/

Dash/U

Litecoin

Maid/U

Monero

Ripple/U

pected shoryou find y% in Mone

ment. Compwer than thect from a the expectedong USD an

d shortfall fo

Exchange ra

Bitcoin/USD

Dash/USD

Litecoin/USD

Maid/USD

Monero/USD

Ripple/USD

for the righ

.tes the 95or cryptocur

nge rate H

/USD

USD

n/USD

USD

o/USD

USD

rtfall is highourself on

ero, you wilparing histoe historical normal disd shortfall fnd short the

for cryptocu

te

D

D

D

ht tail, at the

76

⋅5% expect

rrencies, lef

Historical ES

0.077

0.168

0.135

0.194

0.213

0.180

hest for Monone of thosll actually e

orical ES anES, indicatstribution. for the righcryptocurre

urrencies, rig

Historical Va

0.047

0.083

0.070

0.102

0.103

0.079

e 95% level

0.95ted shortfa

ft tail

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4, No. 1

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Finance andISSN 2

2017, Vol.

d 0.035

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d Banking 2374-2089

4, No. 1

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2017, Vol.

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4, No. 1

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Finance andISSN 2

2017, Vol.

d Banking 2374-2089

4, No. 1


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2017, Vol.

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4, No. 1

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2017, Vol.

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4, No. 1

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2017, Vol.

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4, No. 1

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2017, Vol.

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2017, Vol.

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Finance andISSN 2

2017, Vol.

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d Banking 2374-2089

4, No. 1

248,

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Finance andISSN 2

2017, Vol.

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4, No. 1

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