A Quantitative Analysis of Bitcoins.pdf

University of Groningen

Bachelor’s thesis

A Quantitative Analysis of Bitcoins

Author:T. van den Ende

Supervisor:prof. dr. P.A. Bekker

June 24, 2013

Abstract

In this paper, the correlations between the returns of Bitcoins and returns of real financialassets is under consideration. To that end, 6-hourly data on the exchange rate of Bitcoinsin Euro’s and daily data on the closing yield of German & Spanish 10 year governmentbonds were obtained. First, an overview of the asset under consideration, Bitcoins, andthe techniques used in this paper is given. These techniques comprise of the General Auto-Regressive Conditional Heteroskedasticity method for modelling the time-varying volatilityof financial time series and the Dynamic Conditional Correlation method for modelling thetime-varying correlation between various financial assets. After practising these methods onthe Dow Jones, the government bonds and the DAX index, they are applied to construct dy-namic conditional correlations between the daily returns of Bitcoins and German & Spanish10 year government bonds.

Contents

1 Introduction 21.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Background 42.1 Bitcoins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 DCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 GARCH applied 83.1 Dow & Jones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Bitcoins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 German & Spanish government bonds . . . . . . . . . . . . . . . . . . . . . . . . 9

4 DCC applied 124.1 German 10 year government bonds & the DAX index . . . . . . . . . . . . . . . 124.2 German & Spanish 10 year government bonds . . . . . . . . . . . . . . . . . . . . 13

5 Correlation between Bitcoins & government bonds 145.1 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 DCC Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Conclusion 17

7 figures 187.1 GARCH Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.2 DCC Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1

1 Introduction

In this report, Bitcoins are under consideration. Bitcoins are a relatively new, digital currency,which is on the rise lately and is considered by many to be the next economic bubble. Nev-ertheless, Bitcoins still are increasingly popular, with a current market capitalization of 913.5million Euro[4], and their value, measured by the exchange rate in other currencies, increases aswell. Many consider its independence of the real financial world and regulations, and the lack of(possibly failing) intermediaries as its main advantage and as the foundation for its valuation.These statements are, however, not based on any empirical, quantitative research. Therefore,this report contains an elaborate quantitative research on this subject.

This issue raises many preliminary questions. First of all, what assets are a good representationof the real financial world? And how can the (in)dependence of financial assets be measured?Also, how can the reaction of the returns of Bitcoins on failing intermediaries be measured?Furthermore, in order to be able to provide a meaningful comparative analysis, the return ofBitcoins should be thoroughly investigated.

To be able to provide a sensible analysis on the relation of the return of Bitcoins with fail-ing intermediaries, and as indicators of the real financial world, distinctive European bonds willbe considered. Now that some preliminary questions have been posed, a more general problemcan be formulated.

1.1 Problem Formulation

The preceding introduction entails the following problem, which will be central in this report:

Are the returns of Bitcoins related to the returns of several European government bonds?

To be able to give a quantitative analysis of this problem, the preceding preliminary problemsshould be tackled first. First of all, for the Bitcoins, both the daily and 6-hourly returns of itsexchange rate in Euro’s will be considered, and as indicators of the real financial world, dailyreturns of German & Spanish 10 year government bonds will be considered. The former isdeemed as a relative safe investment, whereas the latter is, nowadays, regarded quite risky, dueto the ongoing financial crisis in Europe, in which Spain was negatively involved. One wouldexpect that this negative involvement is reflected in the evolution of the returns of the Spanish 10year government bonds. Germany, on the contrary, was not a direct instigator of the crisis andtherefore, the returns of the German 10 year government bonds are expected to be imperturbable.

These expectations, as well as the underlying assumptions on the riskiness, will be investigatedby examining the evolution of the (volatility of the) returns of the individual assets. For thelatter, General Auto-Regressive Conditional Heteroskedasticity (GARCH) models will be devel-oped. To fully understand the use of this type of models, the GARCH model will be applied tothe Dow Jones. After its use is demonstrated, GARCH models will be applied to the returns ofthe exchange rate of Bitcoins in Euro’s.

2

Finally, after the individual examination of all these financial assets and some preliminary com-parisons, the cohesion of these assets will be investigated. To that end, Dynamic ConditionalCorrelation (DCC) models will be constructed to estimate correlations. To get a grasp of the useof DCC models, it will first be applied to the DAX index and the German 10 year governmentbonds, of which earlier similar research has been conducted[1], and on the Spanish & German10 year government bonds. After its use is demonstrated, the DCC method can be applied tothe returns of Bitcoins and the returns of the Spanish & German 10 year government bonds,hopefully yielding sensible results.

It is expected that this analysis will show that the returns of the exchange rate of Bitcoinsare uncorrelated with the returns of the German 10 year government bonds, since the returns ofthis bond are likely to be imperturbable, whereas the returns of the exchange rate of Bitcoinsare likely to fluctuate heavily. On the other hand, it is expected that returns of the exchangerate of Bitcoins are correlated with the returns of the Spanish 10 year government bonds, sincethe latter will probably be heavily influenced by the Euro crisis, whereas Bitcoins are regardedas a rewarding alternative to assets influenced by the Euro crisis.

On a final note, both the GARCH and DCC models will be estimated using Sheppard’s MFEToolbox[16].

3

2 Background

2.1 Bitcoins

In this section, a description of Bitcoins is given, and its advantages but also its risks are elab-orated on. Bitcoins are a digital currency, similar to the Euro and Dollar. In essence, Bitcoinsare some sort of encrypted codes, which can be found online. Finding a new Bitcoin can becompared to finding a new prime. It is not difficult to identify the first primes (2,3,5 etc.), butwhen arriving at higher orders, it becomes increasingly difficult to identify the next prime, infact, the computational time needed to find a new prime goes to infinity. Since finding a newBitcoin is that difficult, scarcity is involved, and scarcity is the foundation of the valuation of allgoods. To cope with the increasing computational time, many online so-called ’mining’ groupshave been created, in which the members’ computers are linked in a combined effort to findnew Bitcoins, and since it is possible to split a Bitcoin in numerous parts, the resulting findingsare equally divided amongst its members. Nowadays, new Bitcoins can still be found relativelyeasy, which creates a modern-day gold-fever. But is this new financial product an interestinginvestment or not?

The exchange of Bitcoins is based on a p2p-system (peer to peer). This implies there is nointermediary, such as a bank or a broker involved. Therefore, the safety of one’s investments isnot influenced by the malfunctioning and bankruptcy of banks, which, in the current financialsituation, is considered as an huge advantage. This is the main foundation of the increase in thepopularity of this virtual currency in the past year. Nonetheless, Bitcoins bear some substantialrisks. First of all, Bitcoins are a globally accessible product, without boundaries and thereforeunrelated to governments, hence no regulations have been made yet, which implies no state willprovide in social security for this financial product. In case of a full depreciation, this impliesall investments vanish. Furthermore, Bitcoins are stored in an online wallet, and there havealready been numerous cases in which these online wallets were hacked and emptied, which isessentially pickpocketing. Another downside is the probability of the invention of new, safer,digital currencies, which might lead to a full depreciation of Bitcoins[14].

Is this yet another financial bubble, or is this a serious attempt to create a new, solid cur-rency? Although some webshops and advocates already enabled paying with Bitcoins, the mostimportant distinction with other currencies, is linked to the origin of the need for a nationalcurrency: tax payments. Unlike a government issued currency, Bitcoins cannot be used for thispurpose, and are therefore not fully acknowledged, which will eventually lead to its downfall.

The recent global awareness of Bitcoins led to a huge increase in its value, measured by itsexchange rate with other currencies, however, all negative news, mostly regarding some safetybreach, has quite an impact on Bitcoins as well, which suggests that Bitcoins are very volatile.[2]Nevertheless, little quantitative research has been conducted so far, except for some analyses onthe different type of buyers of Bitcoins[7].

2.2 GARCH

The volatility of asset returns are mainly important to investors since it is a measure for thestability of a financial product. A high volatility implies a higher risk of depreciation, but itmight also lead to higher returns. Either way, the level of uncertainty increases with the volatility.Furthermore, volatilities of different assets can be compared and correlations can be computed,which creates the possibility of hedging a portfolio by selecting various assets which are not or

4

hardly correlated. Hence, asset return volatilities are central to finance, although they are notdirectly observable. Nonetheless, it has been shown that volatilities behave in some characteristicmanner. They, for instance, are prone to clustering. Furthermore, volatilities evolve more or lesscontinuous, and they are subject to the leverage effect, which implies the effect of an increasein returns on the volatility is different from the effect of a decrease in returns[17]. In thisreport, volatilities of several time series will be considered. Financial asset return volatilitiesare time varying. One way to model them, is to create a time series by computing, at timet, the weighted standard deviation of the asset return of some preceding period. To that end,both an exponentially and uniformly weighted framework are constructed, in which the prior 19observations are included, by using the following formulas, where rt is the return of the asset attime t:

(Uniform weight) σ̂t =

√√√√∑19i=0

(rt−i − 1

20

∑19i=0 rt−i

)219

(Exponential weight) σ̂t =

√√√√√ 19∑i=0

e−i/20(rt−i − 1

20

∑19i=0 rt−i

)2∑19

i=0 e−i/20

These estimates do not account for the leptokurtosis of volatilities, hence they do not accu-rately measure the profitability in the tails. In 1982, Engle[8] proposed to model time-varyingconditional variance with Autoregressive Conditional Heteroskedasticity (ARCH), using laggeddisturbances. It appeared that the dynamic behaviour of conditional variance could only beaccounted for when using a high order of lagged variables, giving rise to the Generalized Autore-gressive Conditional Heteroskedasticity (GARCH) model of Bollerslev[6], which is based on aninfinite ARCH specification, reducing the number of estimated parameters to two. GARCH iscapable of registering the clustering (an autoregression effect), but still does not fully account forthe leptokurtosis. It, however, is the most commonly used method for computing the volatilitiesof asset returns. In order to set up the GARCH model, first, the return process rt needs to bedefined:

rt = µt + σtzt

zt ∼ i.i.d.,

E[zt] = 0,

V ar(zt) = 1.

Then the GARCH(p,q) model is defined by:

σ2t = ω +

p∑i=1

αiε2t−i +

q∑j=1

βjσ2t−j .

Here,εt = rt − µt = σtzt.

From these estimated standard deviations of the returns, the annualized estimated volatility iscalculated, where T is the number of observations in one year in our dataset.:

σ̂t,ann = σ̂t√T .

5

The choice of parameters p and q for which the GARCH model has the best fit is based on threecriteria, namely the Log-likelihood criterion, Akaike information criterion (AIC) and the Bayesianinformation criterion (BIC). The Log-likelihood criterion amounts to choosing the model withthe highest Log-likelihood, whereas the AIC and BIC imply choosing the model with the lowestvalue for the AIC and BIC. All three methods for the estimation of σt will be applied to theexchange rate of Bitcoins (in Euro’s) and the 10 year German and Spanish government bonds,but first it is applied on the Dow & Jones index.

2.3 DCC

The GARCH(p,q) model can be applied to a single financial time series, but often a multivariatetime varying volatility model is of interest, since it includes the estimated standard deviationsof the returns of the individual time series as well as the estimated correlation between the re-turns of the financial time series under interest, which, as mentioned before, enables investors tohedge their investments, a method to decrease the probability of default and the size of potentiallosses. In this report, the main purpose for the construction of time varying correlations betweenfinancial assets is to verify whether or not the movements of the exchange rate of Bitcoins can berelated to the developments of other financial product, which, in turn, reflect the developmentsin the entire financial system.

A well-know multivariate model was proposed by Engle, the Dynamic Conditional Correlationmodel[9][15]. The DCC model assumes that the zero-mean returns of n financial time series areconditionally multivariate normal, with expectation zero and a covariance matrix Ht, that is:

rt ∼ N(0, Ht),

Ht = DtRtDt.

Here, Dt is a n× n diagonal matrix with entry (i, i) equal to σi,t, the standard deviations of thereturns of asset i at time t, and Rt is the time varying correlation matrix.

The estimation of a DCC model is two-fold: first, a GARCH(p,q) model has to be fitted toeach of the financial time series individually, yielding an estimate of the standard deviations ofits returns. These estimated standard deviations serve to standardize the return process. Thatis, for each of the n time series under consideration, the return process is transformed into awhite noise process:

rt ∼ N(0, Ht)⇒ zt ∼ N(0, I),

zi,t =ri,tσ̂i,t∀i ∈ 1, .., n.

Then, the DCC(M,N) model is defined by:

Qt =

(1−

M∑m=1

αm −N∑

n=1

βn

)Q̄+

M∑m=1

αm(zt−mz′t−m) +

N∑n=1

βnQt−n.

Here, Q̄ is the unconditional covariance of the standardized residuals resulting from the GARCH(p,q)estimation. Furthermore, let Q∗t be a diagonal n× n matrix with entry (i, i) equal to the squareroot of qi,i, the (i, i)th element of Qt. Now, Rt can be decomposed such that:

Rt = Q∗−1t QtQ∗−1t .

6

Element (i, j) of Rt will be of the form:

qi,j,t√qi,iqj,j

= ρi,j,t.

Another way to model the time varying correlations, is to create a time series by computing, attime t, the weighted correlation between two asset returns during some preceding period. Tothat end, both an exponentially and uniformly weighted framework are constructed, in whichthe current and prior 19 observations are included, by using the following formulas, where zi,t isthe standardized return of asset i at time t, calculated using the same method as for the DCC:

(Uniform weight) ρ̂i,j,t =

∑19k=0 zi,t−kzj,t−k

19

(Exponential weight) ρ̂i,j,t =

19∑k=0

e−k/20zi,t−kzj,t−k∑19k=0 e

−k/20

Throughout this report, a DCC(1,1) model will be applied to the standardized returns of thefinancial assets.

7

3 GARCH applied

3.1 Dow & Jones

For the Dow & Jones Industrial Average, daily closing prices were obtained, from March 11999, until February 28, 2013[12], to demonstrate how the GARCH model works and what itscharacteristics are. A plot of these daily closing prices can be found in Figure 1. From 2002until 2008, the prices of this asset rise steadily and in 2008, there is a considerable drop, whichis likely to be caused by the great financial crisis, after which the Dow Jones Industrial Averagesteadily recovered and currently is at its highest level again. From these prices, the zero-meandaily returns are calculated by applying the following formulas, where T denotes the number ofobservations in the dataset:

rt =Pricet

Pricet−1− 1,

r̄ =

T∑t=1

rtT,

r̃t = rt − r̄.

First of all, the number of lags to include needs to be specified. The estimation of a GARCH(1,2)and GARCH(2,2) model proved to be impossible, but the statistics the estimation of a GARCH(1,1)and GARCH(2,1) model yielded, can be found in Table 1.

GARCH(1,1) GARCH(2,1)Loglikelihood 11035.82 11050.88AIC -3.1631 -3.1669BIC -3.1578 -3.1598

Table 1: Log-likelihood criterion, AIC and BIC of a GARCH(1,1) and GARCH(2,1) modelestimation on the daily returns of the Dow Jones Industrial Average

The GARCH(2,1) model scores best on all three criteria, implying a GARCH(2,1) model fits bestto the data. In Figure 2, a plot of the annualized volatility of the returns of this asset, estimatedby the GARCH(2,1) method, the uniform weighted standard deviation and the exponentialweighted standard deviation can be found. Although the GARCH(2,1) estimate is slightly largerin most cases, the estimates follow the same trends, which suggests the GARCH(2,1) is estimatedcorrectly, hence from here onwards, the GARCH estimates will no longer be compared to thedifferently weighted estimates of the standard deviation. Early 2008, there is a peak in theannualized volatility of the returns, which can be related to the steep decline in the closingprice of the Dow Jones Industrial Average during that period. From here on, only annualizedvolatilities will be considered in this report, unless stated otherwise.

3.2 Bitcoins

For the Bitcoins exchange rate in Euro’s, 6-hourly data was obtained, from July 17, 2010 untilApril 4, 2013[3]. A plot of this exchange rate over time can be found in Figure 3. After January2013, this exchange rate suddenly explodes, resulting in a 500% increase within 3 months. Therewas a lot of media attention for Bitcoins around that period, and it became a worldwide hottopic, which might explain this rigid increase in value. From the exchange rates, the zero-mean6-hourly returns are calculated by applying the formulas stated in the previous section. The

8

estimation of a GARCH(2,2) model proved to be impossible, but the statistics the estimation ofa GARCH(1,1), GARCH(1,2) and GARCH(2,1) model yielded, can be found in Table 2.

GARCH(1,1) GARCH(1,2) GARCH(2,1)Loglikelihood 8258.72 8275.04 8258.72AIC -2.1380 -2.1417 -2.1375BIC -2.1331 -2.1352 -2.1310

Table 2: Log-likelihood criterion, AIC and BIC of a GARCH(1,1), GARCH(1,2) andGARCH(2,2) model estimation on the 6-hourly return of the exchange rate of Bitcoins

The Log-likelihood criterion, AIC and BIC indicate that the GARCH(1,2) model has the bestfit. A plot of the volatility of the 6-hourly returns of this asset, estimated by the GARCH(1,2)method, can be found in Figure 4. Between 2011 and 2012, the exchange rate is rather unstableand in August 2012, there is a temporarily steep rise and decline in the exchange rate, which isreflected well in the estimation of the volatility of the returns by means of quickly alternatingpeaks and drops.

To compare the effect of the number of observations included in the analysis, also daily data forthe same period were obtained. The statistics the estimation of a GARCH(1,1), GARCH(1,2),GARCH(2,1) and GARCH(2,2) model yielded, can be found in Table 3.

GARCH(1,1) GARCH(1,2) GARCH(2,1) GARCH(2,2)Loglikelihood 1314.97 1333.64 1326.21 1333.64AIC -1.3579 -1.3751 -1.3674 -1.3731BIC -1.3427 -1.3549 -1.3472 -1.3478

Table 3: Log-likelihood criterion, AIC and BIC of a GARCH(1,1), GARCH(1,2), GARCH(2,1)and GARCH(2,2) model estimation on the daily return of the exchange rate of Bitcoins

The Log-likelihood criterion, AIC and BIC indicate that the GARCH(1,2) model has the bestfit. In Figure 5, a plot of the estimated volatility of the returns of this asset, based on 6-hourly and daily data, is given for the period from February 22, 2012 until February 21, 2013,to zoom in on the differences. The 6-hourly estimates fluctuate more than the daily estimates,moreover, the daily estimates resemble a moving average of the 6-hourly estimates. This impliesthe 6-hourly estimates contain more information, which is rather intuitive, since the Bitcoins areprone to substantial fluctuations within a 24 hour time period, which the daily estimates neglect.Therefore, the 6-hourly estimates form a better basis, in terms of information, for the followingdiscussion.

3.3 German & Spanish government bonds

For the German 10 year (N) government bond yields with face value(FV) 100 and for the Spanish10 year government bond yields with face value 100, daily closing data were obtained, from Febru-ary 27, 2008, until June 6, 2013 and from February 2, 2006 until June 6, 2013, respectively[11].From these yields (y), the daily return can be computed by applying the following formulas,where T denotes the number of observations in the dataset:

9

Price(t) = FV · e−yN ,

rt =Pricet

Pricet−1− 1,

r̄ =

T∑t=1

rtT,

r̃t = rt − r̄.

In Figure 6, a plot of the prices over time are given. The German 10 year government bond(from here onwards abbreviated to Ger10yr) prices show an upward trend. One explanationcould be that investors tend to resort to government bonds during financial crises, but the Span-ish 10 year government bond (from here onwards abbreviated to Sp10yr) prices do not showthat trend. This might well be caused by the Euro-crisis. Spain faced a large deficit, a growingdebt and a high unemployment rate, which resulted in the severe downgrading of the rating ofSpain by credit-rating agencies from late 2011 until June 2012, leading to the various price dropsand peaks[10]. The Spanish economy has undergone some transformations and some positivenumbers have been published lately[5], leading to a decrease in the yields and an increase in theprices, which is reflected in Figure 6 as well.

For the daily returns of the Ger10yr, the estimation of a GARCH(2,2) model proved to beimpossible, but the statistics the estimation of a GARCH(1,1), GARCH(1,2) and GARCH(2,1)model yielded, can be found in Table 4.


Table 4: Log-likelihood criterion, AIC and BIC of a GARCH(1,1), GARCH(1,2) andGARCH(2,1) model estimation on the daily return of the Ger10yr

The AIC and BIC indicate that the GARCH(1,1) model results in the best fit, whereas the Loglikelihood criterion suggests that the GARCH(1,2) model has the best fit. For the daily returnsof the Sp10yr, the estimation of a GARCH(2,2) model proved to be impossible, but the statisticsthe estimation of a GARCH(1,1), GARCH(1,2) and GARCH(2,1) model yielded, can be foundin Table 5.


Table 5: Log-likelihood criterion, AIC and BIC of a GARCH(1,1), GARCH(1,2) andGARCH(2,1) model estimation on the daily return of the Sp10yr

The Log likelihood criterion suggest that the GARCH(1,2) model is the best fit, whereas theAIC and the BIC indicate that the GARCH(1,1) model results in the best fit. Therefore, theGARCH(1,1) is taken to be the best fit for both the assets.

10

For the Sp10yr & Ger10yr, a plot of the volatility of the daily returns, estimated by theGARCH(1,1) method, can be found in Figure 7. The fluctuations in the yield and prices forthe Sp10yr, caused by the impact of the Euro crisis and the lately improving economic situation,are reflected by a substantial increase in the volatility of the returns, and the major deprecia-tions occur simultaneously with the announcements of the downgrading of the Spanish ratingby credit-rating agencies[13]. The volatility of the return of the Ger10yr fluctuates around aconstant rate, with an occasional peak in late 2011, in the midst of the Euro crisis. In compari-son, the volatilities start harmoniously, but, from late 2009 onwards, suddenly move in opposeddirections. In the next section, these underlying relations are thoroughly investigated, as well asthe relation of the return of Bitcoins with the return on the prices of the Ger10yr and Sp10yr,by means of a Dynamic Conditional Correlation (DCC) model.

11

4 DCC applied

4.1 German 10 year government bonds & the DAX index

First, the DCC model is applied to the returns of the Ger10yr and the DAX, the main stock indexof Germany, of which daily closing prices are obtained from January 02, 2001 until June 12, 2013,which are transformed into daily zero-mean returns using the formulas developed in Section 3.1.The first step of the estimation of the DCC model is standardizing the zero-mean returns, usingstandard deviations estimated by a GARCH(p,q) model. A GARCH(1,1) model has alreadybeen estimated for the Ger10yr returns, resulting in estimates of the standard deviation whichcan be used to standardize the zero-mean returns, as discussed in Section 2.3. For the returnsof the DAX index, the estimation of a GARCH(1,2) & GARCH(2,2) proved to be impossible,but the estimation of a GARCH(1,1) & GARCH(2,1) model yielded the statistics mentioned inTable 6.

GARCH(1,1) GARCH(2,1)Loglikelihood 9219.48 9227.75AIC -2.9083 -2.9103BIC -2.9026 -2.9026

Table 6: Log-likelihood criterion, AIC and BIC of a GARCH(1,1) and GARCH(2,1) modelestimation on the daily return of the DAX index

The criteria indicate that the GARCH(2,1) model is the best fit, and the resulting estimatedstandard deviations of the returns were used to standardize the zero-mean returns. Now thatthe standardized zero-mean returns are established, a DCC(1,1) model can be estimated. Asstated before, the daily closing prices of the DAX index are obtained from January 02, 2001until June 12, 2013. Nonetheless, a DCC model can only be constructed for time series with anequal number of observations, hence only the intersection of time-instances is considered, whichcomprises the daily closing prices of both assets from February 27, 2008, until June 6, 2013.

The estimated time varying correlation between the returns of the DAX index and the Ger10yris plotted in Figure 8. Throughout the whole period, this correlation is negative, starting with avalue of -0.8, after which the correlation revolves around approximately -0.53, with some outliersnear -0.8 and -0.22. One explanation of the negative correlation in this period could be that in-vestors tend to resort to the safer regarded government bonds during financial crises. Therefore,the Ger10yr thrives well during the credit & Euro crisis, whereas the DAX index suffers from it.Nonetheless, in their report, Wong & Vlaar[1] found a positive dynamic correlation from 1988until 1999, indicating that in that period, the explanation mentioned did not have a (decisive)influence.

In Figure 9, the estimated dynamic conditional correlation is plotted against the manually createdconditional correlation, using both an uniform and an exponential framework. These crudely es-timated conditional correlations exceed -1 every now and then. Since a correlation lower than -1is theoretically impossible, these values are not depicted. The estimates move harmoniously, butthe weighted conditional correlations have a higher dispersion and therefore, their outliers areextremer. This is an indication that the DCC estimates are constructed in a proper way, hencefrom here onwards, the DCC estimates will no longer be compared to the differently weightedestimates of the correlation.

12

4.2 German & Spanish 10 year government bonds

In Section 3.3, a GARCH(1,1) model has already been estimated for both the Ger10yr and Sp10yrreturns, resulting in estimates of the standard deviations which can be used to standardize thezero-mean returns, as discussed in Section 2.3. With these standardized returns, a DCC(1,1)model can be estimated. The assets have different datasets, consisting of a different amountof observations, whereas a DCC model can only be constructed for time series with an equalnumber of observations. To cope with these differences, the whole dataset of the Ger10yr andonly those observations of the Sp10yr that occur at the same time as the observations of theGer10yr are included, which comprises the daily standardized zero-mean returns from February27, 2008, until June 6, 2013. The prices of the bonds, depicted in Figure 6, and the volatilitiesof the returns of the bonds, depicted in Figure 7, suggest that until March 2010, the returns ofthe bonds have a positive correlation, and from March 2010 until March 2013, the bonds have anegative correlation, with its most negative value around the July, 2011.

In Figure 10, the estimated time varying conditional correlation between the daily returns ofthe Ger10yr and Sp10yr is plotted. This estimated conditional correlation exceeds 1 occasion-ally, which is theoretically impossible, hence these values are omitted. The correlation reflects theassumptions based on Figure 7 quite well, that is, first there is a positive correlation, and indeedfrom March 2010 onwards, the correlation attains negative values, for which a possible causewas already given, namely the Euro crisis, since in the mid-2010, the Bank of Spain announcedan estimate of the amount of potential troubled loan exposure, which considerably exceeded theloss the Spanish banking system was able to cope with[10]. Slow recovery periods, in which thecorrelation between the returns approaches and passes zero, are followed by steep drawbacks,which is reflected by the movements of the prices.

In this section two DCC models have been established and analysed to demonstrate its use,and in the following section the DCC method is applied to the return of the exchange rate ofBitcoins and the return of the Spanish & German 10 year government bond.

13

5 Correlation between Bitcoins & government bonds

Prior to this section, applications of the GARCH and DCC model have been demonstrated. Inthis section, the gained experience with these models is used to return to the core of this research.In Sections 3.2 and 3.3, a preliminary research on the 6-hourly returns of the exchange rate ofBitcoins and the daily returns of the Sp10yr & Ger10yr has already been conducted, a firstobservation on the differences in the individual volatility of the returns of the bonds has alreadybeen made and a dynamic conditional correlation between the return of the bonds has beenestimated. Here, the DCC method will be applied to obtain dynamic conditional correlations ofthe daily returns of the exchange rate of Bitcoins and of the returns of the Sp10yr & Ger10yr,but first a more elaborate hypothesis on these correlations is formulated.

5.1 Hypothesis

In the problem formulation, a superficial hypothesis regarding the potential correlation betweenthe daily returns of the exchange rate of Bitcoins and the daily returns of the Sp10yr & theGer10yr has already been posed. In the previous sections, these assets were investigated, whichenables a more thorough, adequate hypothesis regarding the potential correlation.

The initial assumption of a correlation between the daily returns of Bitcoins and Sp10yr wasbased on the following reasoning. A property of Bitcoins is that it is not linked to any regula-tion, and no intermediary is needed. Therefore, Bitcoins are not prone to ’bad’ performancesof the banking system or the European system as a whole, which is considered an advantage.This is also the foundation for its increased popularity, since Bitcoins are an alternative for the’failing’ (government) bonds. This suggests that a decrease in the returns of Sp10yr, which areconsidered sensitive to the performances of the European (banking) system, will lead to an in-crease in the returns of Bitcoins, which implies there is a negative correlation between the dailyreturns of the exchange rate of Bitcoins in Euro’s and the daily returns of the price of the Sp10yr.

Figure 3 indicates that the price of Bitcoins increases up until the first half of 2011, decreasesthe second half of 2011 and slowly increases from 2012 onwards, every now and then disruptedby temporarily drawbacks. Figure 6 indicates that the price of the Sp10yr decreases until theend of 2010, alternates between peaks and drawback from there until 2012, decreases the firsthalf of 2012 and increases thereafter. These opposite movements suggest a negative correlationas well. Therefore, a negative correlation is expected.

In the problem formulation, it was stated that the correlation between the daily returns of theGer10yr and the Bitcoins are expected to be zero. This was based on the presumed imperturba-bility of the returns of the Ger10yr on the one hand, despite the credit & Euro crises, and thepresumed periodically alternating positive and negative returns of the Bitcoins on the other hand.

The observed non-diminishing upward trend of the price of the Ger10yr in the time periodunder investigation strengthens the assumption that, contrary to the Sp10yr, the returns of theGer10yr are not negatively influenced by the Euro crisis, but it rescinds the original assumptionof imperturbability. The returns of the Ger10yr appear to thrive on the crisis instead, which,based on the preceding reasoning, also holds for the returns of Bitcoins. That, combined withthe observation that the exchange rate of the Bitcoins in Euro’s shows an every now and thendisrupted upward trend from 2012 onwards, leads to an adapted hypothesis, namely a positivecorrelation.

14

5.2 DCC Analysis

In the previous section, the daily standardized zero-mean returns of the Ger10yr and Sp10yrhave already been constructed. In Section 3.2, a GARCH(1,2) model has been estimated for the6-hourly return of the exchange rate of Bitcoins, resulting in estimates of the standard deviations.In order to construct the daily standardized zero-mean returns of the exchange rate of Bitcoins,the 6-hourly returns and the estimated standard deviations of the 6-hourly zero-mean returnsof the exchange rate of Bitcoins need to be transformed into daily estimates. Therefore, the(estimated standard deviations of the) observed returns within a day are accumulated for everyday, using the following formulas:

rt,daily =

(Tt∏i=1

(rt,i + 1)

)− 1,

σ̂t,daily =

√√√√ Tt∑i=1

σ̂2t,i.

Here, Tt denotes the number of 6-hourly observations during day t, which is 4 most of the time,but occasionally an observation is missing. From these transformed data, the daily zero-meanreturns can be computed, using the formulas developed in Section 3.1. It appears that thistransformation has little impact on the estimates, and therefore a full analysis of this impact isomitted. From these daily zero-mean returns and estimated daily standard deviations of the re-turn, the daily standardized zero-mean returns of the exchange rate of Bitcoins are constructed,using the formulas developed in Section 2.3.

As stated before, the number of observations of the time series under considerations should bealigned to enable the estimation of a DCC model, hence only the intersection of time-instancesis considered, which comprises the daily standardized zero-mean returns of the three assets fromJuly 17, 2010 until April 4, 2013. The dynamic conditional correlations between the daily re-turns of the assets under investigation, estimated by a DCC(1,1) model, are depicted in Figure 11.

5.3 Results

Nearly all estimated conditional correlations between the daily return of the exchange rates ofBitcoins and the daily return of the Ger10yr attain negative values, with a mean value of -.0257and nearly all estimated conditional correlations between the daily return of the exchange ratesof Bitcoins and the daily return of the Sp10yr attain positive values, with a mean value of 0.0368.This observation indicates that the daily return of the exchange rate of Bitcoins does not standalone, that is, there is some cohesion, although little, between the evolution of this return andthe real financial world, which is a remarkable result.

Nonetheless, this cohesion with the financial assets chosen in this report deviates from the statedexpectations. Where a negative correlation with the daily return of the Sp10yr and a positivecorrelation with the daily return of the Ger10yr was presumed, the opposite proves to be true.

Several explanations for this observation can be given. One might argue that both Bitcoinsand the Sp10yr are regarded as risky assets, whereas the German government bonds are deemedmore safely. In financial crises, this would imply that investors resort to the Ger10yr instead of

15

the Sp10yr and Bitcoins, leading to the observed correlations. There are plenty other, equallysound possible causes for the observed correlations. However, these results refute the most citedexplanation for the popularity of Bitcoins, its independence of the European banking systemand its current bad performances, since, apparently, a decrease in the return of the Sp10yr, anindication of the aggravation of the Euro crisis, does not result in an increase in the return ofthe exchange rate of Bitcoins.

The correlation between the daily return of the exchange rate of Bitcoins and the daily returnof the Ger10yr starts around -0.1 but has an upward trend, whereas the correlation between thedaily return of the exchange rate of Bitcoins and the daily return of the Sp10yr starts around 0.1,then fluctuates between 0 and 0.05 for a while, after which it has a diminishing trend. Moreover,both conditional correlations seem to converge to zero in the time period under investigation.It might well be that, in the near future, the correlations switch sign, or that the return of theexchange rate of Bitcoins becomes independent of the financial assets under consideration, whichwould imply that the correlations will continue to converge to zero. A future research similar tothis one might be able to expose the true development of the correlations.

16

6 Conclusion

In this report, the main focus was on finding potential correlations between the returns of Bit-coins and returns of real financial assets. To that end, 6-hourly data on the exchange rate ofBitcoins in Euro’s and daily data on the closing yield of German & Spanish 10 year governmentbonds were obtained and the GARCH & DCC methods were applied.

First of all, non zero correlations were found, and the observation that almost all of them attainnegative/positive values throughout the time period under investigation, respectively, indicatesthat there actually is a negative/positive effect between the financial assets under consideration,although little. Nevertheless, both correlations tend to zero over time, hence, a future researchmight elaborate on the significance of the observed correlations.

Although the recent popularity of Bitcoins is often attributed to its independence of the Eu-ropean/international banking system, this research has shown that the daily return of Bitcoinsis positively correlated with the daily return of Spanish 10 year government bonds, and nega-tively correlated with the daily return of German 10 year government bonds, which contradictsthe assumptions that Bitcoins are bought more often as a reaction to the Euro crisis.

Moreover, the observations indicate that Bitcoins do not behave independent from the realfinancial world at all, in fact, there is a clear distinction between the relation of Bitcoins withpresumed volatile assets, and the relation of Bitcoins with assets that are deemed trustworthy,that is, the returns of the exchange rate of Bitcoins are positively connected to the returns ofthe volatile asset under consideration, the Spanish 10 year government bonds, and negativelyconnected to the returns of the safe asset under consideration, the German 10 year governmentbonds.

Finally, a future research might investigate the relation of these correlations over time and mightclarify whether the correlations converge to zero, switch sign, or develop another trend.

17

7 figures

7.1 GARCH Analysis

Dow Jones

01−Jan−1999 01−Jan−2002 01−Jan−2005 01−Jan−2008 01−Jan−2011 01−Jan−20140.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5x 10

4

Time

Price

Figure 1: Daily closing prices of the Dow Jones Industrial Average, from March 1 1999, untilFebruary 28, 2013

18

01−Jan−1999 01−Jan−2002 01−Jan−2005 01−Jan−2008 01−Jan−2011 01−Jan−20140

0.2

0.4

0.6

0.8

1

1.2

1.4

Time

Estim

atedvolatility

σ̂GARCH

σ̂UNI

σ̂EXP

Figure 2: Annualized volatility of the daily return of the Dow Jones Industrial Average, estimatedby a GARCH(1,1) model and both a uniform & exponential framework, from March 2 1999, untilFebruary 28, 2013

19

Bitcoins

01−Jul−2010 01−Jan−2011 01−Jul−2011 01−Jan−2012 01−Jul−2012 01−Jan−2013 01−Jul−20130

50

100

150

Time

Exch

an

ge

ra

te

Figure 3: 6-hourly exchange rate of Bitcoins in Euro’s, from July 17, 2010 until April 4, 2013

20

01−Jul−2010 01−Jan−2011 01−Jul−2011 01−Jan−2012 01−Jul−2012 01−Jan−2013 01−Jul−20130

2

4

6

8

10

12

14

Time

Estim

ate

d v

ola

tilit

y

Figure 4: Annualized volatility of the 6-hourly return of the exchange rate of Bitcoins, estimatedby a GARCH(1,2) model, from July 17, 2010 until April 4, 2013

21

Jan 12 Apr 12 Jul 12 Oct 12 Jan 13 Apr 130

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time

EStimatedvolatility

σ̂daily

σ̂6−hourly

Figure 5: Annualized volatilities of the 6-hourly and daily returns of the exchange rate of Bitcoins,estimated by GARCH(1,2) models, from 22 February, 2012 until 21 February, 2013

22

German & Spanish 10 year government bonds

01−Jan−2008 01−Jan−2009 01−Jan−2010 01−Jan−2011 01−Jan−2012 01−Jan−2013 01−Jan−201445

50

55

60

65

70

75

80

85

90

Time

Price

Spain 10 yrGerman 10 yr

Figure 6: Daily closing prices of German & Spanish 10 year government bonds from January 1,2008 until June 6, 2013

23

01−Jan−2008 01−Jan−2009 01−Jan−2010 01−Jan−2011 01−Jan−2012 01−Jan−2013 01−Jan−20140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time

Estim

atedvolatility

σ̂Spain10yr

σ̂German10yr

Figure 7: Annualized volatilities of the daily returns of the German & Spanish 10 year governmentbonds, estimated by GARCH(1,1) models, from January 1, 2008 until June 6, 2013

24

7.2 DCC Analysis

01−Jan−2008 01−Jan−2009 01−Jan−2010 01−Jan−2011 01−Jan−2012 01−Jan−2013−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

Estim

atedcorrelation

Time

Figure 8: Correlation between the daily returns of the DAX index & German 10 year governmentbonds, estimated by a DCC(1,1) model, from February 27, 2008, until June 6, 2013

25

01−Jan−2008 01−Jan−2009 01−Jan−2010 01−Jan−2011 01−Jan−2012 01−Jan−2013−1

−0.5

0

0.5

Estim

atedCorrelation

Time

ρ̂uniform

ρ̂exponential

ρ̂dcc

Figure 9: Correlation between the daily returns of the DAX index & German 10 year governmentbonds, estimated by an uniform/exponential framework & a DCC(1,1) model, from February 27,2008, until June 6, 2013

26

01−Jan−2008 01−Jan−2009 01−Jan−2010 01−Jan−2011 01−Jan−2012 01−Jan−2013−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Estim

atedcorrelation

Time

Figure 10: Correlation between the daily returns of the Spanish & German 10 year governmentbonds, estimated by a DCC(1,1) model, from February 27, 2008, until June 6, 2013

27

01−Jul−2010 01−Jan−2011 01−Jul−2011 01−Jan−2012 01−Jul−2012 01−Jan−2013−0.1

−0.05

0

0.05

0.1

0.15

Estim

atedcorrelation

Time

ρ̂Sp,BTC

ρ̂Ger,BTC

Figure 11: Correlation between the daily returns of the exchange rate of Bitcoins and the dailyreturns of the Spanish & German 10 year government bonds, estimated by a DCC(1,1) model,from July 17, 2010 until April 4, 2013

References

[1] P. Vlaar A. Wong. Modelling time-varying correlations of financial markets. Dutch CentralBank (Research Department), WO Research Memoranda 739, 2003.

[2] European Central Bank. Virtual currency schemes. October 2012.

[3] Bitcoincharts. Historical data on the exchange rate of bitcoins. http://bitcoincharts.

com/.

[4] Blockchain. Data on bitcoins. http://www.blockchain.info.

[5] Bloomberg. Spain’s crisis fades as exports lead the way. http://www.bloomberg.com/news/2013-06-03/spain-s-crisis-fades-as-exports-lead-the-way.html.

[6] T. Bollerslev. Generalized autoregressive conditional heteroskedasticity. Journal of Econo-metrics, 31:307–327, 1986.

[7] A. Shamir D. Ron. Quantitative analysis of the full bitcoin transaction graph. WeizmannInstitute of Science, 2012.

28

[8] R. Engle. Autoregressive conditional heteroskedasticity with estimates of the variance ofunited kingdom inflation. Econometrica, 50:987–1007, 1982.

[9] R. Engle. Dynamic conditional correlation - a simple class of multivariate garch models.Journal of Business and Economic Statistics, 20:339–350, 2002.

[10] C. Harrington. The spanish financial crisis. Master’s thesis, University of Iowa, 2011.

[11] Investing. Historical data on government bonds. http://nl.investing.com/

rates-bonds/.

[12] CBS News. Historical data on the dow jones index. http://markets.cbsnews.com/

cbsnews/.

[13] CBS News. Information on the euro crisis. http://markets.cbsnews.com/cbsnews/.

[14] P. Piasecki. Design and security analysis of bitcoin infrastructure using application deployedon google apps engine. Master’s thesis, Wydzia Fizyki Technicznej, 2012.

[15] K. Sheppard R. Engle. Theoretical and empirical properties of dynamic conditional cor-relation multivariate garch. National Bureau of Economic Research, Working Paper 8554,2001.

[16] K. Sheppard. Oxford mfe toolbox. http://www.kevinsheppard.com/wiki/MFE_Toolbox,2009.

[17] R.S. Tsay. Analysis of Financial Times Series. John Wiley & Sons, 2005.

29

Documents

A Quantitative Analysis of Bitcoins.pdf