i

CAN A UK INVESTOR GAIN DIVERSIFICATION BENEFITS AND REDUCE

PORTFOLIO RISK USING ALTERNATIVE ASSETS INCLUDING BITCOIN? AN

EMPIRICAL ANALYSIS USING TESTS OF MEAN-VARIANCE SPANNING.

University of Strathclyde

Tom Alexander Nutton

201121183

Submitted in partial fulfilment of the requirements for the degree of BA (Hons) Business

Enterprise and Finance

ii

Abstract

The purpose of the study is to determine whether a UK investor with a conservative

investment approach and a correspondingly conservative asset portfolio can benefit from

incorporating a variety of alternative assets, including Bitcoin within their portfolio holdings.

This research examines asset performance data for the period July 2010 to January 2015.

This period is post Global Financial Crisis (GFC) and also post introduction of the

cryptocurrency Bitcoin, an alternative asset of central interest in this research.

Following the framework described by Kan and Zhou (2012) for the assets concerned, asset

performance data is analysed using a statistical framework to assess the differences between

two mean-variance frontiers. The mean variance frontier technique tests whether alternative

assets improve portfolio performance in terms of returns earned over the period in question.

The technique also examines portfolio diversification by calculating the degree of correlation

between the different asset types in the portfolio.

Portfolio risk management is an important risk management technique in relation to

investment practice. The alternative assets used in this research are studied in order to

calculate which assets hold the greatest potential benefits for institutional investors. The

assets are examined in terms of both the returns that could be earned and the increased

diversification that could be achieved, by incorporation of the new assets in turn into the

traditional portfolio.

From the test assets considered the research findings confirm that Bitcoin outperforms all

other test assets over the time series evaluated. Bitcoin would therefore have been a

beneficial holding in terms of additional returns and improved diversification benefits. This

paper also examines Bitcoin, both in relation to the asset itself including how it works and its

likely appeal to institutional investors.

iii

Declaration

This dissertation is submitted in partial fulfilment of the requirements for the Degree

Bachelor of Arts in the University of Strathclyde, and accords with the University

regulations for the programme as detailed in the University Calendar.

I declare that this document embodies the results of my own work and that it has been

composed by myself. Following normal academic conventions, I have made due

acknowledgement of the work of others.

Signed

Date

iv

Acknowledgements

I would very much like to thank my family and friends for their encouragement, guidance

and inspiration throughout the process of writing this dissertation and through all of my

years at university. I must also focus my gratitude towards my supervisor Juliane Thamm

and Professor Johnathan Fletcher for providing me with invaluable advice and guidance

which motivated me to write this dissertation to the best of my ability.

v

Table of Contents

Page Number

1.

Introduction 1

2. Literature Review 4

2.1. Introduction 4

2.2. Portfolio Theory 4

2.3. Portfolio Diversification 5

2.3.1. Number of Securities 6

2.3.2. Correlation and Diversification 6

2.3.3. Internationalisation and Diversification 7

2.4. Asset Types and Allocation 8

2.4.1. Alternative Assets 9

2.4.2. Oil 10

2.4.3. Gold 10

2.4.4. Real Estate 11

2.4.5. Treatment of Alternative Assets 11

2.5. Bitcoin as an Alternative Asset 12

2.6. Mean-variance spanning 17

2.6.1. Bitcoin and Mean-variance Spanning 20

3. Data and Methodology 23

3.1. Introduction 23

3.2. Previous Studies 25

3.3. Benchmark Portfolio (K) 26

3.3.1. Descriptive Statistics – Benchmark Portfolio (K) 27

3.4. Alternative Assets - Test Assets (N) 27

3.4.1. Descriptive Statistics - Test Assets (N) 29

3.5. Description of Correlation Matrices 30

3.6. Methodology 31

3.6.1. Huberman and Kandel (1987) Methodology and Formula 31

3.6.2.

Kan and Zhou Step-down Application 33

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Page Number

4. Empirical Results and Discussion 35

4.1. Introduction 35

4.2. Gold 36

4.3. Brent Oil 37

4.4. Euro 39

4.5. Japanese Yen 39

4.6. Emerging Market Bond Index (EMGBi) 41

4.7. Bitcoin 42

4.8. All Test Assets in the Benchmark Portfolio 43

4.8.1. Short Selling 45

4.9. Limitations and Other Observations 46

4.10.

Summary of Empirical Results 48

5. Conclusion 50

5.1. Introduction and Summary of Research 50

5.2. The Future Professional Investor 51

5.3. Diversification Benefits 52

5.4. Bitcoin and Future Entrepreneurial Developments 53

5.5.

Suggestions for Further Research 53

6.

Bibliography 55

7. Appendices 65

Appendix 1 - Other Methods of Mean-variance spanning 67

Appendix 2 - Bitcoin – Technical 68

Appendix 3 - Risks with Bitcoin 68

Appendix 4 - Mining and Blockchain Technology 70

Appendix 5 - Bitcoin Exchanges 71

Appendix 6 - Cyber, Regulatory, Political and Ethical Implications of Bitcoin 71

Appendix 7 - Other Cryptocurrencies 72

Appendix 8 - Behavioural Finance and Bitcoin

73

8.

Glossary 75

9.

Tables 77

10. Figures 84

1

1. Introduction

The Investment Management industry is hugely important to the UK and world economies.

Dakers (2014) conveys that the total funds under management in the UK alone were

estimated at GBP 5.0 trillion in 2014. Pro-active risk management of asset portfolios is

therefore key in terms of both anticipating and responding to the risks that such investment

portfolios face. Portfolio risks arise in a number of ways in terms of fluctuation in asset

values caused by global economic events, fluctuations which in turn are driven by investor

sentiment. Diversification of the assets within portfolios is a key risk mitigation technique of

central importance to professional investment practitioners and academics alike. Specifically,

Fragkiskos (2013) confirms that developing and understanding new ways of reducing risk,

while optimising portfolio returns, has held the interest of academics for decades and will

continue to do so owing to its global importance.

As the world becomes ever more connected in this digital age, one by-product of inter-

connectedness is that different types of investable assets, as detailed by Bernstein and

Pinkernell (2007), steadily become more correlated with each other in terms of performance.

The negative aspects of this correlation were demonstrated by the Global Financial Crisis of

2007-9, in terms of the knock-on effects in one economy quickly spreading throughout the

globe. There is an interesting analogy here between financial markets and the global

climate: the so-called butterfly effect where small effects could potentially trigger another

catastrophic climatic event owing to interconnectedness. The inherent risk within the

connected dynamic global economic system means it is of paramount importance for

investors to evaluate new pro-active ways of managing risk, in order to try and minimise the

down-side consequences of a future catastrophic event. Risk management techniques in the

context of investment portfolios include minimising risk by diversification, in order to stay

ahead of the tendency for different assets to become more connected and more correlated

with each other over time. The goal is to reduce risk by limiting correlation and therefore

reducing the impact of losses when catastrophic global events occur.

In terms of investor attitude towards different types of investable assets, Taylor (2012)

expresses that many practising professional portfolio managers will be from the so-called

baby-boomer generation with birth years between the end of World War II and 1965. The

latter part of the cohort now being in their 50s and many professionals this age will be in

senior management positions in industry and commerce. This boomer generation grew up

well before the advent of the connected digital age, but in terms of demographics they will

remain the driving force in boardrooms and investment committees for some years to come.

2

When considering the profile of a typical UK-based investor, their portfolio construction and

the portfolio managers, Damato (2012) details that baby-boomers are likely to be in control

of most key decisions on risk, reward, asset weightings and portfolio mix. Yet, this

generation of professionals charged with preserving capital and providing returns in excess

of risk-free rates, may according to Durden (2013), lack the insight, appetite, techniques and

communication networks to fully understand and exploit rapid developments in the new

digital age. This constraint could also preclude investors from considering the new breed of

digital investable assets. Hence if this constraint were to exist in terms of portfolio risk

management, it may mean that potential new methods of portfolio diversification could be

inadvertently over-looked, or even if considered, deemed too alternative to receive active

consideration for portfolio inclusion at the present time.

One example of recent ground-breaking digital developments is the creation of virtual

currencies known as cryptocurrencies; these are entirely new digital products that can be

used for transferring value between two parties to a financial transaction in a secure manner.

According to Yueh (2014) because cryptocurrencies are not controlled or issued by any

central authority, this presents an unconventional challenge to embedded and established

financial systems and instruments. As an inherent part of their design, cryptocurrencies

create a market demand, which in turn sets their value – a value which fluctuates over time

as do other investable assets. The cryptocurrency reviewed in this research - Bitcoin – has

taken the virtual world by storm, its appeal accelerated by the always-connected millennial

generation.

This work therefore investigates the practice of diversifying investment portfolios by

utilising alternative asset-types. There are many potential asset-types that could have been

used as alternatives; examples would include securitised financial instruments such as

catastrophe bonds and longevity derivatives. However, the alternatives actually selected are

chosen for their risk profile, data availability and investor appeal. The alternative assets

therefore include commodities, currencies, emerging market bonds and cryptocurrencies.

Specifically with the latter – Bitcoin - which is the most well-established in this new digital

asset class. Bitcoin has grown rapidly as a new asset class and has also featured in the press

following a number of high-profile incidents and has become the asset that can no longer be

ignored.

This work’s contribution is via research and data analysis, to present evidence to suggest that

Bitcoin does hold potential for investors to utilise as a new asset and should be included

within investment portfolios. Bitcoin allocations added to the reference portfolio do improve

3

diversification and mitigate risk exposure. Extending the foundations laid by previous

papers on this subject allows a comprehensive identification, treatment and understanding of

where the value of Bitcoin lies in portfolio risk enhancement, via its value and associated

returns being uncorrelated with more traditional investment asset-types.

The dissertation is structured to firstly, review the key literature relating to portfolio theory.

Then, associated risk management and optimisation techniques are critiqued in order to

highlight and present areas relevant to further investigation and research. From this

foundation the research and methodology are outlined, developed and discussed in detail.

Following this, the results of an empirical analysis of actual asset performance are presented.

A discussion of the findings considers the implications, potential for application to

investment portfolio risk management practices and the resulting benefits to portfolios in

terms of greater capital security and enhanced value. Finally, the concluding chapters

highlight the conclusions that can be drawn from this work and also detail recommendations

for follow-up and future studies in this subject area. The work also makes extensive use of

appendices and uses these for some of the background technical details relevant to

cryptocurrencies, result tables from data analysis and graphical representation of output

including the mean-variance frontiers plotted for the various asset combinations studied.

4

2. Literature Review

2.1. Introduction

Risk management techniques available in respect of investment portfolios are of critical

importance to commercial, academic and economic audiences. In addition to diversification

and allocation, other active portfolio management methods include short-selling, stop-loss

and take-profits strategies. Such techniques are important tools to manage the risks inherent

within an investment portfolio. In order to develop a solid grounding on the subject of

portfolio risk management, this literature review will firstly detail the principal portfolio

theories, and then describe how risk management practice can be further enhanced using

alternative assets, such as Bitcoin, within a risk management approach. The work

concentrates on the main risk mitigation measures of diversification and allocation, to

enhance understanding and assess implications upon the future direction of portfolio

management. From here the foundations of the empirical analysis technique - mean-variance

spanning – are introduced and discussed in terms of its applications and overall

effectiveness. Finally, recent academic studies reviewing Bitcoin’s diversification potential

will be explored in order to position this work in terms of its value and contribution to the

subject of portfolio risk management.

2.2. Portfolio Theory

From investment and academic perspectives, Markowitz (1952) laid the foundations of

portfolio theory by applying mathematical principles in a financial context, creating a

framework for investors to utilise in order to minimise risk while maximising their expected

return. Today, it still remains highly regarded as the principal theory underpinning portfolio

selection and optimisation. Specifically, the Markowitz (1952) model defines a number of

distinct assumptions regarding investor behaviour within financial markets. These are:

investors seek efficient mean-variance levels for their portfolios; investors only have a one-

period horizon; investors are risk-averse and markets are fully transparent and open.

An extension to this foundation is the Capital Asset Pricing Model (CAPM) theory -

established by Sharpe (1964) and Lintner (1965) – which devises an asset-pricing model

which represents the market. Fama and French (2004) explain that this model assumes that

risk and return have a linear relationship, and investors are able to borrow at a risk-free (rf)

rate (see Glossary 2) and all investors commonly agree on the distribution of expected

5

returns. Best and Hlouskova (2000) clarify that by combining the rf asset and a set of risky

assets an investor is able to devise a mean-variance portfolio which sits as a tangent on the

efficient frontier. Although this model has stood robustly for decades, it fails to fully address

and account for all market irregularities; inspiring the emergence of behavioural finance as a

discipline and revisions to the CAPM, in order to explain observed market and investor

behaviours. Further studies question whether levels of diversification are dependent upon the

individual investors’ risk appetite, rationality and financial understanding. More specifically,

scholars believe that not all investors follow the concrete assumptions proposed by the

CAPM. Again, this delves into progressive theories from the field of behavioural finance

(see Appendix 8 for further details). One such market anomaly is identified by Coeurdacier

and Guibau (2011) who describe that investors have a strong tendency to suffer from home

and familiarity biases. This observed propensity leads to investment portfolios that are over-

weight in domestic-based assets and investments.

Further extensions of portfolio theory explored by Lintner (1965) and Beja (1972)

established a refined understanding of the fundamentals of diversification, revealing that

portfolios are subject to both unsystematic and systematic risks. Samuelson (1967) states that

investors will tend to diversify with identically-distributed risks, even when selected from

independent asset pools.

Due to the simplicity and success of these portfolio theories and principles, further studies

have applied the fundamentals to investigate portfolio performance across a broad range of

markets and asset classes. De Roon and Nijman (2001) present a practical extension - mean-

variance spanning – and how it can be used to review the addition of supplementary assets

into a predetermined asset portfolio. This area of research will be evaluated in the latter

stages of this chapter.

2.3. Portfolio Diversification

There is a body of research documenting the benefits of diversification upon enhancing

portfolio performance with regard to risk and return, such as Bloomfield (1977) and Statman

(1987). Although these studies universally agree that diversification is an essential

component of any portfolio, academics largely disagree on the number and variety of asset

allocations required to achieve optimal levels of diversification. Fundamentally, holding less

correlated assets in a portfolio lowers the overall vulnerability of a portfolio to risk. Scholars

such as Bernstein and Pinkernell (2007) also suggest that the modern financial landscape is

6

becoming increasingly correlated, thereby steadily increasing risk over time. I discuss these

issues further in the latter stages of the review.

2.3.1. Number of Securities

The first highly debated area of diversification concerns the number of securities needed to

attain optimal diversification benefits. Evans and Archer (1968) were the first to establish

that the portfolio variance and the number of securities held demonstrate an inverse

relationship. They found that by investing in 10 randomly selected securities they were able

to fully replicate market volatility. This study revealed that portfolios are subject to both

unsystematic and systematic risk. Essentially, systematic risk (commonly referred to as

market risk) cannot be diversified away. Unsystematic risk can be diversified away by

adding more uncorrelated assets until no further material risk reduction can be achieved by

this technique. Alternatively, Bloomfield, et al. (1977, p. 25) emphasise that portfolios

consisting of 20 stocks “attain a large fraction of the total benefits of diversification”.

Statman (1987) on the other hand insists that portfolios should consist of upwards of 30

stocks, yet in practical terms the benefits of portfolio diversification significantly decelerate

after 10 stocks. More recent empirical evidence from Campbell et al. (2001) indicates that

from 1963 to 1985 investment portfolios could be effectively diversified using 20 stocks, yet

from 1986 to 1997 some 50 stocks were required to achieve the same level of unsystematic

risk reduction. Along similar lines, Sankaran and Patil (1999) were also able to deduce that

portfolios consisting of greater numbers of securities can achieve higher Sharpe ratios (see

Glossary 3), although the marginal diversification benefits progressively decelerate after 10

securities. Similarly, de Vassal (2001) evaluated the performance of portfolios with

increasing numbers of securities: by using returns from the Russell 1,000, over the period

1992 to 1999; he was able to devise random portfolios which spanned the range of 3 to 100

stocks. He documented that larger portfolio sizes demonstrate progressively lower levels of

variance and inherent risk, thus confirming the finding of Evans and Archer (1968).

2.3.2. Correlation and Diversification

Another widely researched area, which has a direct influence upon the effectiveness of

diversification, is the correlation between assets and financial markets. For example, Lessard

(1973) explores the diversification benefits of investing in a range of equity markets by

7

reviewing correlation levels; finding that significant levels of correlation are detrimental to

portfolio performance. Taking this into consideration, Odier and Solnik (1993) and Longin

and Solnik (1995) were able to conclude that correlations between U.S. and international

stocks were significantly higher in falling market conditions, which leads to an avoidance of

investing abroad in times of crisis. Similarly, Ang and Chen (2001) analyse weekly portfolio

returns from July 1963 to 1998 and demonstrate that during bull markets, correlations are

notably lower than during normal economic conditions, whereas bear market conditions

demonstrate much higher levels of correlation. Essentially, these findings contradict the

assumptions that all returns strictly follow normal distributions. Therefore it can be inferred

that the diversification benefits will be overestimated during bull markets and

underestimated during bearish episodes. Hence, this development may have significant

implications for portfolio composition, which is potentially of greatest economic significance

during periods of instability. These results indicate that investors need to be fully aware of

underlying market conditions when devising and actively managing portfolios.

Most notably, Bernstein and Pinkernell (2007) evaluate the diversification benefits among 11

alternative asset classes. They document that the correlations between assets classes and

market indices generally increased over time, thus lowering the potential of diversification as

a risk mitigant. Fundamentally, they were able to deduce that the correlation co-efficient in

isolation provides a poor indication of the level and effectiveness of diversification benefits.

Similarly, research by Statman (2007, pp.3) reflected that correlations within portfolio

optimisation procedures are “incomplete indicators of the benefits of diversification”.

Therefore Carrieri et al. (2007) proposed that correlations should be considered in tandem

with standard deviations, as well as using the techniques such as mean-variance spanning, to

allow for a full market integration measurement.

2.3.3. Internationalisation and Diversification

Another growing area of academic research is the internationalisation of financial markets

and its implications for portfolio management. Essentially, the explosion of globalisation in

recent years has made diversification more flexible, and essential to mitigate unsystematic

risk levels.

For instance, Levy and Sarnat (1970) and Solnik (1974) demonstrate that international

portfolio allocations can significantly improve Sharpe Ratios (1966) and diversification

benefits. Similarly, Driessen and Laeven (2007) analyse monthly returns from 1985 to 2002

8

from 52 countries to gauge whether international diversification is still beneficial in more

recent economic conditions. To review the economic and statistical significance of their

findings they undertake mean-variance spanning tests and document the difference in Sharpe

ratios between local and internationally-orientated funds. Specifically, they detail that a local

investor can reap substantial diversification benefits when considering investment in

international markets. These findings are also mirrored by Fletcher and Marshall (2005) who

used a revised Bayesian methodology to evaluate how a UK-based investor can benefit from

international diversification. They also find that Sharpe ratios (1966) are significantly

enhanced by introducing foreign assets into UK-denominated portfolios.

However, Christofferson et al. (2010) found that the increasing levels of globalisation have

increased the correlation between markets. This development is also evident through

research from Eiling and Gerard (2007), who stress the importance of exploring new

alternative assets and developing economies for enhancing investment portfolios. Erb et al.

(1995) suggest that investors are however susceptible to significant levels of additional risk

in international markets, such as credit, political and regulatory risk.

Further academic research from Bolgar (2012) suggests that times of economic instability

have a substantial effect upon the behaviour of investors and portfolio management. Before

the Global Financial Crisis (GFC) investors had faith in dividing portfolios with 60%

allocated to high risk/return (growth) assets such as equity and 40% assigned to safe assets,

namely government bonds. However, according to Peter (2015), due to the increasing

correlation between interest rates, oil prices, inflation and marketable assets, traditional

portfolios now struggle to hedge against underlying risk using these techniques.

2.4. Asset Types and Allocation

Brinson et al. (1995) establishes that conventionally, investors construct portfolios consisting

of equity, government bonds and global currencies. Specifically, they highlight that equities

in general provide investors with ample flexibility to achieve high levels of diversification.

Ciner et al. (2013) on the other hand finds robust evidence to suggest that investors hold

government and corporate bonds as a key hedging instrument for stock market volatility:

although the empirical research collated by the authors only estimated this for UK and US

datasets.

9

Likewise, global currencies are also highly regarded portfolio allocations. For instance,

Marion (2010) establishes that the Japanese Yen is deemed to be a safe-haven currency in

times of economic uncertainty owing to the perceived conservative policies of the Japanese

government. This effect causes large amounts of capital to flow predominantly to the

Japanese Yen, Swiss Franc and the U.S. Dollar in uncertain economic times. Similarly,

Stubbington (2014) finds that the Euro is showing some signs of becoming a safe haven

currency. Not only do these global currencies have consistent records in maintaining value,

research from Joy (2011) highlights that the U.S. Dollar has demonstrated an increasingly

negative relationship with gold, which makes it a strong hedge against gold price movements

and vice versa. On the contrary, Eun and Resnick (1988) provide empirical evidence to

suggest that exchange rate uncertainties can significantly affect international portfolios,

meaning currencies have an underlying level of unsystematic risk which is difficult to

diversify away.

2.4.1. Alternative Assets

The scholars Amin and Kat (2003), Chen et al. (2002), Chen et al. (2005) and Anson (2006)

all claim that traditional portfolio holdings have limited diversifying qualities. They

emphasise, however, that alternative assets and alternative investments as a whole,

demonstrate superior performance, both in insolation and when included within portfolios

comprising traditional asset classes.

Importantly, when evaluating alternative asset holdings, Chen et al. (2002) stress that it is

crucial to understand the underlying characteristics of each asset class when considering

them for investment purposes. Similarly, Yau et al. (2007) characterise the groupings as the

following: traditional alternative assets which consists of Real Estate, Private Equity and

Commodities. The second classification is modern alternative investments which comprise

of managed futures, hedge funds and distressed securities. Categorically, Bitcoin does not

neatly fit into any of these sub-sets due to its inherent distinctiveness. However, it could be

inferred that due to Bitcoin’s highly volatile and irrational nature, it could fit within the

distressed asset category.

More recently, Tang and Xiong (2010) find that investing in commodities, among other

alternative assets, is becoming progressively more widespread and popular within the most

developed and developing nations. On similar lines, Fabozzi et al. (2008) found that

commodity futures significantly improved the mean-variance efficiency of traditional

10

investor portfolios denominated in world equities, bonds and risk-free assets. Moreover,

findings from Kat (2006), Kat and Oomen (2007) and Gorton and Rouwenhorst (2006)

further indicate that commodities have specific qualities which enhance the diversification

efficiency of traditional-based portfolios. Although Cheug and Miu (2010) point out that the

supposed diversification benefits of commodities only hold significance in the long-run.

2.4.2. Oil

Beioley (2015) establishes as an alternative investment oil holds great significance within the

global financial market. Studies from Jones and Kaul (1996) document U.S. and Canadian

stock prices were correlated in line with oil price developments, hence indicating a

correlation to some degree which may limit its diversification potential. However, Arouri

and Nguyen (2010) refute these findings, indicating that stock and oil markets are weakly

correlated and generally move independently of the respective sector activities. Hence, an

allocation of oil leads to portfolios with superior performance and improves a portfolio’s

risk-return characteristics. Further findings from Arouri and Nguyen (2010, pp.4537) reveal

that a portfolio with “10% invested in Brent crude oil, the average weekly return increases

from 0.523% to 0.618%, while the standard deviation decreases from 3.180% to 3.143%”.

Notably, Arouri and Nguyen (2010) make the assumption that the investor’s risk preference

follows a strictly concave function, which favours risk-aversion when interpreting the

results.

2.4.3. Gold

Gold is another commodity often considered for investment portfolios. Specifically, Bauer et

al. (2010) advise that gold is deemed to be a respectable hedge and a safe financial haven

during periods of financial distress. However, they found that this only seems to be the case

for stocks over the short-term, and gold is not a safe haven for bonds when evaluating UK,

German and U.S markets during times of economic turbulence.

On similar lines, Erb and Harvey (2013, p.40) highlight that gold is an attractive asset due to

its “low correlations with other tradable assets”. Saad (2012) agrees confirming that 30% of

the financial analysts surveyed in 2011 portrayed gold to be the greatest long-term

investment for portfolios. Besides this, De Long (2011) documents that when devising

11

portfolios, investors are generally faced with an opportunity cost, as gold is expensive to

possess when interest rates are high and significantly cheaper when rates are low.

2.4.4. Real Estate

As the UK equity market has demonstrated poor performance in recent years, this has

increased the attractiveness of including real estate investments within mixed-asset

portfolios. Studies such as Chan et al. (2011) have established that real estate investments are

highly effective for portfolios, especially over the long term as they act as an inflation hedge.

Also the inherently low correlation of real estate with financial assets can enhance the

benefits of portfolio diversification. For instance, Lee and Stevenson (2006, pp.10) identified

that the “position of real estate changes across the efficient frontier from its return enhancing

ability to its risk reducing facility”. Therefore, Lee and Stevenson (2006) were able to

categorise real estate as an effective diversifying option, rather than a high-yield addition to

mixed asset portfolios. Notably, this study was conducted at the height of the property

market before the onset of the 2007 global financial crisis. Conversely, in light of this recent

financial crisis, Lizieri (2013) portrays that real estate is in fact significantly correlated to

financial assets during periods of economic instability, thus demonstrating that real estate is

only an effective diversification option during periods of prosperity and stability.

2.4.5. Treatment of Alternative Assets

As identified by Kat (2006), investors tend to evaluate and treat alternative investments in

the same manner as large capitalisation stocks and government bonds, i.e. investors tend to

prefer lower risk assets. This tendency may lead investors to overweight their portfolios with

alternative asset-types resulting in excessive risks if they do not fully understand the

underlying characteristics of these assets and what determines their value. Therefore it can

be inferred that limited knowledge regarding alternative asset classes may lead potential

investors to rely more upon speculation. Similarly, Baker and Filbeck (2013) state that

alternative assets are inherently challenging to assess due to their individual complexities

and the difficulty of constructing comparable performance benchmarks. Furthermore, they

also state that limited sources, academic coverage and relevant data available may cause

issues in analysing the assets. In short, Baker and Filbeck (2013, pp.4) suggest that some

12

investments are “unavailable or unsuitable for the general public due to their complexity or

structure”.

Before an investor considers the option of adding an alternative asset into a given portfolio,

Greer (2000) explains that such alternative assets need to satisfy various suitability criteria.

Firstly, the asset should heighten the expected utility of the portfolio, hence a higher return

for a given level of risk (i.e. Sharpe ratio). Secondly, the expected returns from this risky

asset cannot be reproduced simply by differing the combination and weights of the different

assets already held in the reference portfolio.

This research demonstrates that investors are cautious and need to be highly informed when

dealing with new alternative asset classes within portfolio construction. This should be taken

into account when considering the inherent risks and diversification benefits of the latest

financial innovation, Bitcoin and may explain why Bitcoin has not found favour in portfolios

thus far.

As Bitcoin is the main focus of this research, the following section will establish how

Bitcoin can be used for portfolio diversification and why it should be of interest to

academics. Previous literature has hinted that Bitcoin has investment potential, yet these

sources fall short in grasping a complete empirical understanding. The literature review to

follow will therefore consider the current position of academic research on Bitcoin.

2.5. Bitcoin as an Alternative Asset

Bitcoin exists purely in the digital electronic environment of cyber-space. However, even

without a form of physical presence, it is worthy of consideration as an investable asset.

Bitcoin is a new type of financial instrument and a mechanism for transferring economic

value from one party to another party without the need for a trusted third party. Essentially,

the genius in the underlying Bitcoin technology integrates modern cryptographic techniques

and a de-centralised peer-to-peer infrastructure to create a purely digital payment medium.

Bitcoin’s technology allows the effective transfers of currency, seamlessly, anonymously

and instantly, without the influence and reliance upon financial intermediaries. Hence, this

new breed of cryptocurrency holds the potential to massively reshape the global relationship

and treatment of money and financial instruments in general.

Bitcoin’s inception in the midst of the Financial Crisis during 2008 has been hailed as a

technological phenomenon which holds the potential to revolutionise the way monetary

13

transactions are conducted. Not only do individual investors have growing interest in

Bitcoin, central banks such as the Bank of England (2014) and financial analysts such as

Blundell-Wignall (2014), Masters (2014) and Grover (2014) have begun to take an interest

in Bitcoin.

Bitcoin was introduced by a secretive programmer, identifying himself under the fictitious

identity of Satoshi Nakamoto (2008). Famously Nakamoto (2008, pp.1) cited in his white

paper that Bitcoin is “an electronic payment system based on cryptographic proof instead of

trust”. Although, sceptics such as Barber et al. (2012) and Moore (2013) contend that

Nakamoto collaborated with other technical academics such as Chaum (1983), Back (2002),

Camenisch et al. (2005), Canard and Gouget, (2007), Dai (1998) and Okamoto (1995).

Nevertheless, Bitcoin is worthy of considering for portfolio inclusion because of its unique

nature and due to the likelihood of its diversifying and risk reducing properties. These unique

properties of Bitcoin have initiated widespread debate within the financial and academic

world. Specifically, economists are puzzled whether to treat Bitcoin as a commodity, a

currency, or even a completely new classification altogether.

Bitcoin differs from traditional currencies for a number of fundamental reasons. Firstly, the

value of Bitcoin is freely determined by market demand, and it is not issued by any central-

controlling institution. Secondly, the Bitcoin supply increases at a predefined decreasing rate

up until 2140 whereby a maximum of 21 million Bitcoins will be materialised, therefore

oversupply will never become a problem. Although Williams (2014) believes that up to 4%

of Bitcoins have already been permanently erased from the system due to issues concerning

hacking, disc-drive failure and fraudulent activities. Market participants have the option to,

much like gold, ‘mine’ Bitcoins to bring them into existence within the economy (further

technical coverage can be found in Appendix 2). These characteristics make Bitcoin an

investable portfolio allocation.

There are a number of conflicting views regarding Bitcoin and to date, studies have

evaluated the significance, impact and value Bitcoin can bring to the global marketplace

from a variety of perspectives. Studies range from technical cryptographic studies, to

political, regulatory and the sociological implications of this innovation. Specifically,

Kristoufek (2013), Barber et al. (2012) and van Wijk (2013) detail that as Bitcoin is still in

its infancy, especially in comparison to gold, there are a number of important factors to

consider. For instance, Stevens (2013) proposed that Bitcoin should be regarded and treated

as just another foreign currency within the global economy. Alternatively, Toma (2012)

classifies Bitcoin as an ‘Electronic Money System’, facilitating mobile network payments.

14

Conversely, Selgin (2013) delves into existing economic and financial frameworks, allowing

him to deduce that Bitcoin is a form of ‘Synthetic Commodity Money’. However, all of these

definitions largely oversimplify the fundamentals of Bitcoin and fail to provide empirical

support for their claims. However, other scholars such as Yermack (2013, pp.1) argue that

Bitcoin emulates similar behaviour to a “speculative investment [rather] than like a

currency” simply as Bitcoins are inherently volatile and have negligible correlation with

other currencies.

On the other hand, Clinch (2013) and HMRC (2014) take an alternative position that Bitcoin

should be treated as a form of ‘private money’. This classification is more promising and

provides some evidence to suggest that Bitcoin has the potential to become a form of

investment. The IRS (2014) rejects all other academic perspectives and raises the prospect

of treating Bitcoin as a unique asset classification altogether. This controversial statement

further deepens the complexity of academic debate on where Bitcoin lies within the financial

spectrum, and rightly so, as Bitcoin’s innovation is revolutionary.

Chong and Wang (2014) investigate the factors that determine the Bitcoin exchange rate.

They find that economic factors such as inflation rates, employment rates and GDP levels in

the U.S. poorly represent Bitcoin exchange rate behaviour. By devising a revised regression-

based test, they were able to deduce that Bitcoin’s price is significantly correlated to

technological factors (mining technology) and investor attention (Google Trends) variables.

These results provide evidence to suggest that Bitcoin behaves independently of economic

conditions and stimuli, unlike other highly traded financial instruments and commodities.

Further research from Brown (2014, pp.1) gauges the level of efficiency and liquidity of the

Bitcoin market, finding that return predictability is “statistically significant [over the] short-

horizon”.

Experts such as Liu (2013) even stretch this scrutiny to the extent of comparing the nature of

Bitcoin to that of a new gold standard; economists have deliberated that Bitcoin shares

qualities similar to gold. Some of these shared characteristics are the scarce supply, the

increasing difficulty of sourcing and materialising; investors have sentimental attachment to

holding them, both are used for trading, and finally, both are generally uncorrelated to

traditional investments. Thus leading Li et al. (2014, p.14) to establish that the process of

mining Bitcoins is “similar to the production of gold”. Furthermore, Shafiee (2010) indicates

that the mining of Bitcoin holds parallels with gold mining, as essentially, supply and

demand – as well as the cost of mining - has a direct relation to gold price. All of these

distinct characteristics add to Bitcoin’s credibility and potential for investment purposes.

15

Wu and Pandey (2014) examine the functionality of Bitcoin being adopted as a legitimate

currency and then further investigate whether Bitcoin is a powerful investment option. First

of all, they test whether Bitcoin functions effectively as: a medium of exchange, store of

value and a unit of account. From this research, they find that Bitcoin address two of the

criteria yet fails to act as a reliable storage of value due to its liability of newness and

extreme volatility. Taking this into consideration, they extend their analysis by conducting a

variety of performance tests upon a range of investment portfolios. By comparing Sharpe,

Omega and Sortino ratios, as well as the Black and Litterman (1992) approach, they were

able to deduce that even a pessimistic investor could benefit from incorporating Bitcoin

within an investment portfolio. These comprehensive tests give a rounded understanding of

Bitcoin and how investors with varying risk preferences can benefit from Bitcoin within their

investment portfolios.

Although pre-existing economic tests neatly gauge the fundamentals of more traditional

assets and currencies, they fall short in determining and extracting any valuable conclusions

or significant results for cryptocurrencies. In effect, this has stalled academic progression.

Each of these contrasting views hinders academic advancement in understanding Bitcoin.

However, what can be inferred is that Bitcoin does possess the qualities to be a recognised

form of investment.

Perhaps the most profound characteristic of Bitcoin is that it holds no fundamental value as

such – this has led to a number of implications. Essentially, the price of Bitcoin is not pegged

to any form of currency and is determined solely by supply and demand dynamics, i.e. when

demand rises, price rises accordingly. This demonstrates that Bitcoin’s potential intrinsic

value is unbounded and unlimited, as prices are driven by investor sentiment, trust and

market developments within the context of finite supply and an increasingly difficult mining

process.

Typically investors and economists are able to price securities and assets by evaluating

growth, dividends and cash flow forecasts; however, Weisenthal (2013) highlights that

Bitcoin has no intrinsic value. Grinberg (2011) stresses that this lack of ability for investors

to calculate an underlying fundamental value for Bitcoin makes it highly susceptible to

bubble scares and speculation. Fox (2013) conveys that economists and financial analysts

have not encountered financial instruments such as Bitcoin before, causing opinions and

treatments to vary significantly at extreme ends of the spectrum. For this reason, investors

have labelled Bitcoin an extremely high risk investment. Institutional and typical investors

alike have therefore demonstrated a general rejection of considering Bitcoin as the next great

16

financial opportunity. Not only this, but as Bitcoin displays instability (see Appendix 3) it is

deemed undiversified making it a wild-card for inclusion within portfolio allocations.

However, Bitcoin has acted as the catalyst cryptocurrency, spurring on a new wave of

cryptocurrencies and Blockchain (see Appendix 4) innovations. For example Litecoin and

Dogecoin are two other spin-off prototypes that have proven popular, gaining traction and

credibility by streamlining the existing Bitcoin technology (see Appendix 7). In essence,

Sidechain (see Appendix 4) developments will allow for open and continuous innovation

within the Bitcoin universe – allowing more rational investors to get involved from specific

angles. Moreover, Timms (2014) is confident that Blockchain technology can substantially

reduce the cost and time to distribute financial instruments and assets.

There are specific challenges to considering Bitcoin for portfolio allocation. Lee (2013)

suggests that the misunderstood and complex nature of the cryptographic technologies

underpinning Bitcoin has led to a lack of research. This is accentuated further as Xin and

Wang (2014) demonstrate there is generally a lack of empirical understanding of the

underlying dynamics of the Bitcoin infrastructure and make-up. Similarly, researchers such

as Velde (2013) establish that Bitcoin in terms of trading volume is insignificant in relation

to other developed assets and wider economies. Moreover, across all of the reviewed

literature, studies generally lack in robustness due to the relatively small datasets used and

due to the inflexible, standardised econometric and statistical analyses conducted.

Essentially, standard economic and financial performance models perform poorly in

measuring Bitcoin’s significance.

Alternative and high risk assets require an underlying foundation of knowledge to attain an

understanding of their dynamics. This therefore implies that technical innovations such as

Bitcoin, are generally deemed beyond the scope of generic risk-averse investors. However,

this does not rule out institutional or well-informed investors who have the expertise to

understand the investment implications of Bitcoin. Cryptocurrencies in general and Bitcoin

in particular, are new considerations in the context of investment portfolio construction and

diversification and may require specific and possibly new methodologies to investigate them

fully. Conclusive research into the potential diversification benefits of Bitcoin may also help

overcome investors’ reluctance to consider this alternative investment, or at the very least

settle academic debate as to the exact nature of Bitcoin as an asset.

As previously mentioned, De Roon and Nijman (2001) convey that it is of great academic

interest to review if the addition of supplementary assets, into a predetermined asset

17

portfolio, enhances investment opportunities. One such method of evaluating alternative

assets is known as mean-variance spanning.

2.6. Mean-variance Spanning

From a portfolio analysis perspective, Kan and Zhou (2001) demonstrate that it is of high

significance and relevance to understand whether adding additional risky assets into an

existing portfolio will improve mean-variance efficiency. The term ‘risky assets’ refers to a

new set of assets that may have a higher inherent risk in terms of return, volatility and

correlation to the investor’s pool of existing assets. Traditionally, investors evaluate mean-

variance relationships to determine the performance of portfolios, by comparing Sharpe

ratios (Sharpe, 1966) or other performance measures, yet these approaches have limited

application and usefulness.

One frequently used test, which informs investors about the benefits of introducing

additional assets to a benchmark portfolio, was first introduced by Huberman and Kandel

(1987). Essentially, they used a regression-based framework to evaluate and compare the

mean-variance relationship of two portfolios, by using statistical and mathematical

techniques derived from financial theory. Specifically, this mean-variance analysis extends

reasoning from financial theories such as the Capital Asset Pricing Model proposed by

Sharpe (1964) and Lintner (1965). The Huberman and Kandel (1987) test builds upon

theories developed by Merton (1972) and Rolls (1977) who demonstrate that if investors

have a choice between two portfolios, they will demonstrate a tendency to prefer either the

tangency portfolio (i.e. maximises the Sharpe ratio) or the global minimum variance

portfolio (i.e. the portfolio that presents minimum risk for a certain level of return).

Essentially, if spanning occurs, there are no risk or return benefits when introducing the test

assets into the original (or benchmark) portfolio.

Jobson and Korkie (1989) develop the methodology of Huberman and Kandel (1987) further

by establishing mean-variance spanning tests where risk free (rf) assets exist and when they

do not. They conclude that the original Huberman and Kandel (1987) spanning tests should

be compared in conjunction to their empirical approach in order to confidently validate the

acceptance or rejection of hypotheses.

Essentially, mean-variance spanning tests compare the mean-variance frontier of a set of

benchmark assets (K), in relation to another set of benchmark (K) assets that also include

18

additional (N) risky or test assets (K + N). If spanning is present, this demonstrates that

there is no supplementary benefit in incorporating the test asset within the benchmark

portfolio. In other words, this test demonstrates that if the mean-variance frontier of the

benchmark portfolio plus additional assets overlaps with the original (benchmark) portfolio,

then spanning is present. De Santis (1995) demonstrates that a mean-variance frontier

investor cannot benefit from including additional risky assets to the optimal portfolio.

Conversely, if the mean-variance frontier of the original (benchmark) portfolio and the

frontier of the newly constructed portfolio have a singular common point, this demonstrates

and is referred to as intersection – therefore the investor can benefit from addition of the new

risky asset.

The foundations laid by Huberman and Kandel (1987) have inspired further extensions and

variations of empirical tests. There are examples of academic applications of mean-variance

spanning tests in a variety of contexts. For instance, De Santis (1995) and Cumby and Glen

(1990) test whether US investors can enhance portfolio performance by considering

international diversification. Similarly, De Santis (1994), Bekaert and Urias (1996), Errunza

et al. (1999) and De Roon et al. (2001) consider if mean-variance portfolio characteristics are

improved by incorporating assets relating to emerging and other international markets. De

Roon et al. (2001) extend their findings further by identifying that when more test assets are

included within regression-based mean-variance spanning tests, the predictive power of the

test results becomes less significant and less robust.

Furthermore, De Roon and Nijman (2001) demonstrate that Jensen’s alpha (1968) - a

measure used to test the significance of expected returns - and the Sharpe ratio (1966) are

interrelated when considering the covariance of error terms and Jensen’s alpha to define the

possible Sharpe ratios. In simple terms, as the null hypothesis defined by the spanning test

denotes the Jensen’s Alpha as zero, thus implying that the Sharpe ratio should also equate to

zero. Glen and Jorion (1993) investigated the prospect of introducing currency futures into a

well-diversified portfolio consisting of international stocks and bonds. However, Bekaert and

Urias (1996) provide robust evidence to suggest that these aforementioned studies fail to

account for realistic scenarios, as they largely disregard frictional transactional costs, low

liquidity levels, investment constraints and the economic effects of international boundaries.

The academic research conducted to date has therefore comprehensively addressed the issues

surrounding the internationalisation of financial assets and investments. However, these

studies also narrowly focus their attention on futures, international stocks and bond indices;

largely overlooking the prospect of reviewing alternative assets and investments. These

19

studies also appear to make the assumption that international investments are easily

accessible, cost-effective and highly transparent, when in reality the opposite may be true.

Furthermore, the majority of the studies take the perspective and risk appetite of a U.S.

investor, limiting the value of the research. Reviewing the mean-variance spanning literature

has highlighted that this area of research is growing in significance, especially with respect

to internationalisation; yet alternative assets and investments remain largely under-

researched. A possible reason for this is that further development of the spanning

methodology allowing easier assessment of alternative assets classes is only very recent.

It is evident that there are a number of implications regarding the methodologies used in

previous research. Firstly, previous studies tend to concentrate on outdated datasets;

incorporating datasets without careful consideration of market dynamics and underlying

factors. The dated model proposed by Huberman and Kandel (1987) also has some empirical

deficiencies; specifically, their methodology tests the ‘α’ and ‘δ’ jointly and places heavy

weights on delta in hypothesis, therefore leading to biased and skewed results. Although the

most popular mean-variance spanning tests may elegantly capture significant results for old

datasets, they may fall short in accurately characterising developments within modern

datasets. They may also lack relevance when dealing with new investment classes such as

cryptocurrencies. It is essential to extend studies to have a more up-to-date and modern

consideration of new breeds of investment-grade assets. It is therefore appropriate to

undertake revised empirical methodologies that address these issues going forward. The

description and empirical limitations of other mean-variance methods such as the Huberman

and Kandel (1987) approach, among others, are discussed in Appendix 1.

Kan and Zhou (2012) offer a revised extension of the Huberman and Kandel (1987)

regression-based mean-variance spanning framework, allowing an enhanced empirical

approach to test the mean-variance spanning hypotheses. Essentially, they test whether a

U.S. investor holding a portfolio consisting of a 30-Year US Treasury bond and the S&P500

index, can reap diversification benefits from investing internationally. They conclude that an

investor can benefit significantly from diversifying their portfolio through international

markets, although these benefits have decreased over time.

The value in the step-down approach is that it equips the investor with a much broader brief

as to whether to invest in a set of test assets and incorporate them within investment portfolio

(K). This sequential test helps the investor to identify where the benefits of investing are, and

where the rejection of the proposed hypotheses derives from. Most notably, Kan and Zhou

(2012) detail that calculating the power of a mean-variance spanning test using previous

20

methods is inherently difficult to interpret when the test asset (N) is greater than 1; with the

step-down approach, this limitation is rectified.

According to Switzer and Fan (2007, pp.107), the value in the Kan and Zhou (2012) step-

down approach is the fact that the “spanning test examines the two components of the

spanning hypothesis individually and jointly”. Namely, Kan and Zhou (2001) demonstrate

that this revised approach isolates each component of the hypothesis in order to measure the

impact the test asset has upon the portfolio tangency, and secondly, the global mean-variance

of the efficient frontier. In other words, their enhanced method equips the reviewer with the

ability to weight the two components of the mean-variance spanning tests ‘α’ and ‘δ’ (this is

explained in the Methodology section in greater detail) individually, with respect to their

underlying significance to the reviewer. Conducting the test in this fashion helps solve the

power issues associated with the original Huberman and Kandel (1987) tests and provides

significantly more information to the investor regarding portfolio allocation.

From the aforementioned issues with other available methods, it is therefore appropriate to

undertake the Kan and Zhou (2012) step-down approach in this study in order to derive more

informative conclusions. Furthermore, the step-down methodology proposed by Kan and

Zhou (2012) has not been used to evaluate alternative asset allocations, having been

overlooked by previous studies of Bitcoin’s suitability for inclusion within investment

portfolios. These niche areas are where the value and contribution of my research lies.

2.6.1. Bitcoin and Mean-variance Spanning

To date, two studies examine Bitcoin’s suitability for investment purposes, specifically

focusing on its suitability for diversification and portfolio enhancement.

The leading paper on the investment potential of Bitcoin is Brière et al. (2013) investigating

Bitcoin’s performance as a portfolio enhancer. The authors construct a range of conventional

portfolio performance analyses in conjunction with a basic series of spanning tests. The two

portfolios created for these tests consist of traditional and alternative assets, and test the

efficiency of the portfolio when Bitcoin is incorporated and when left out. They found that

Bitcoin is uncorrelated with most assets, and only demonstrates a slight correlation with gold

and inflation-linked bonds. Even though this study uses a dataset which includes Bitcoin’s

extreme price increase and greatest price volatility, the researchers were able to deduce that a

small allocation of Bitcoins can significantly enhance an investment portfolio.

21

This founding paper only utilises two types of simple mean-variance spanning tests, the

original test by Huberman and Kandel (1987) and the somewhat revised Ferson, Foerster and

Keim (1993) test, which presents a general approach on the assumptions that

homoscedasticity and normality are relaxed. Although it is beneficial to have this mean-

variance spanning comparison, the resultant statistics are by no means informative as a

whole, as they only indicate that diversification benefits exist with an allocation of Bitcoin.

Furthermore Brière et al. (2013) give no premise as to where their deductions derive from

and also fail to investigate and highlight where bias in their analysis may be present.

Building upon these findings, Chowdhury (2014) utilises a more up-to-date and

representative dataset, which includes significant price developments and further extend the

spanning tests of the portfolios by incorporating more informative and complete research by

Daskalaki and Skiadopoulos (2011). From this analysis, Chowdhury (2014) was able to

deduce that Bitcoins should be more widely adopted as a diversifying tool, albeit treated with

added caution. To an extent this revised and enhanced spanning test approach addresses a

number of the common empirical downfalls as mentioned in discussion of the earlier

spanning tests. They also provide evidence to suggest that the Markowitz (1952)

assumptions are subject to multiple deficiencies, which may not accurately reflect the actual

gains from investment due to reliance upon two weak conventions. These assumptions are:

asset returns are normally distributed and investor preference follows a standard utility

function.

In a realistic setting, these assumptions are not expected to hold true to a significant degree.

There is sufficient empirical evidence to suggest that asset returns do not follow normal

distributions, especially over short-term horizons; shown for example for a variety of assets

types such as stocks (Peiro, 1999), and commodity futures (Gorton and Rouwenhorst, 2006),

(Kat and Oomen, 2007). Furthermore, Jondeau and Rockinger (2006) elaborate that non-

normality is not accounted for in the optimal portfolio creation. This tendency leads

investors to prefer positive skewness and dislike significant levels of kurtosis, which in turn

leads to utility losses when optimising portfolios.

These studies only visually inspect the mean-variance efficient frontier measurements to re-

inforce their findings, however, to derive more robust results, analysis needs to be conducted

within a statistical framework. The sole test conducted only visualises the changes of mean-

variance efficiency when Bitcoin is included and excluded from a portfolio. Although this

may provide some interesting results, they are highly predictable. As has been highlighted

from the works of Statman (1987), adding more assets into a portfolio will reduce

22

unsystematic portfolio risk accordingly. A more informative methodology should compare

Bitcoin’s performance against other risky assets while ensuring the number of benchmark

assets remains constant; this in turn will produce far richer, comparable and empirically

sound results.

Relatively few academic studies actually consider Bitcoin and conclude it to be a valid

investment vehicle. Although research has been undertaken to investigate Bitcoin’s effectiveness

in portfolio diversification, these studies have various underlying flaws in their approaches and

the implications of their findings lack impact. Moreover, as highlighted in the literature, studies

considering the investment potential of Bitcoin have exclusively taken the perspective of a U.S

investor, leaving the rest of the international financial markets largely untested in this context.

Reviewing the available literature has also established that mean-variance spanning is an

effective way to measure the impact and effectiveness, in terms of risk mitigation, of adding

supplementary assets into an established portfolio. The most comprehensive and informative

mean-variance spanning test of those reviewed – the Kan and Zhou (2012) step-down approach

has not been fully utilised to date by other researchers in this field. Hence the value and

contribution of this work lies in the fact Bitcoin will be analysed from the perspective of the Kan

and Zhou (2012) step-down approach.

Having considered the above literature it is evident that the null and alternative hypothesis

for my research can be defined as follows:

H0: Including alternative assets (and Bitcoin) within a conservative UK portfolio does not

increase mean-variance efficiency and Sharpe ratios.

H1: Including alternative assets (and Bitcoin) within a conservative UK portfolio increases

mean-variance efficiency and Sharpe ratios.

23

3. Data and Methodology

3.1. Introduction

This chapter will outline the datasets that have been used in this empirical analysis, in order

to review the performance of a portfolio of benchmark assets and a series of test assets – the

alternative assets. The method used in this work also uses the Kan and Zhou (2012) step-

down approach to evaluate the impact of adding the test assets. Specifically, the constituents

of the benchmark portfolio and the separate test assets will be detailed in turn and the

respective performance data time-series will be explained: this is the data used in the

analysis.

The data sample used for the following tests consists of 235 weekly returns set out as a time-

series for the period July 2010 to January 2015. It is data for a range of traditional and

alternative investments sourced from both DataStream and Bitcoin-orientated websites. This

time series of data in terms of start-dates and end-dates was selected to incorporate the

launch, introduction and the volatile price developments of the cryptocurrency, Bitcoin, and

also to omit the highly significant and volatile Global Financial Crisis period of 2007-2009.

Today, Bitcoin’s price has shown some signs of maturity, yet there is a long road ahead until

solid conclusions about the cryptocurrency can be established. As of 1st February 2015

Bitcoin price stood at US $214; this is the lowest price point since September 2013.

Chart 1 - Bitcoin Market Price

Chart 1: The above chart presents Bitcoin’s price development over the period 2010 to 2015.

(Bitcoin: Market Price USD, 2015)

24

Since Bitcoin’s inception in 2010, annualised volatility stands at 155%, which is more than

10 times higher than the FTSE All-Share index volatility over of the same period. Sivy

(2013) explains these excessive levels of volatility derive from its undeveloped nature,

infamous associations and speculation-driven price. The most significant deviations in

Bitcoin’s price were fuelled by the Cyprus crisis during March 2013 (Rushe, 2013) and the

collapse of the largest Bitcoin exchange Mt. Gox. (see Appendix 5).

Chart 2 – Bitcoin: Number of Daily Transactions

The above graph demonstrates that the number of Bitcoins traded daily has increased

steadily since July 2012. Notably, trading demonstrates significant periods of volatility,

especially during times of high levels of media attention and speculation – this also has an

adverse effect upon market capitalisation levels (see Appendix 7).

Other research evaluating Bitcoin for investment portfolios use data-sets which analyse

Bitcoin during periods of excessive volatility (December 2013 to May 2014);

understandably, this will amplify the likelihood of biased and unrepresentative results. To

mitigate this risk, I incorporate far more stable data in the analysis as volatility from the

period June 2014 onwards has calmed significantly, thus increasing the chances of the tests

representing Bitcoin in a more robust manner. In the empirical tests, the sub periods of Jan

2013 – Jan 2014 and Jan 2014 to Jan 2015 will also be evaluated separately. The additional

Chart 2: The above chart presents the number of Bitcoin transactions per day over the period 2010-15

(Bitcoin: Number of Transactions per day, 2015).

25

granularity arising from the sub-periods study may help to identify and compare the relevant

factors useful to investors, in terms of how they could construct their portfolios with

inclusion of alternative assets.

3.2. Previous Studies

Studies such as Brière et al. (2013), Wu and Pandey (2014) and Chowdhury (2014) construct

their test portfolios predominantly of traditional assets from the perspective of a risk-neutral

U.S. investor, such as: stocks, bonds and hard currencies.

The key difference between the above-noted studies and this work is that the benchmark

portfolio in this work is geared towards recreating a passive, ill-informed and conservative

UK investor. This hypothetical investor is generally unaware of the technical benefits of

diversification and acts with inherent bias for familiar traditional asset classes and

allocations; therefore heightening the tendency to select benchmarks that are UK-orientated.

My analysis is far more comprehensive and informative for investors than the previous

papers, as this work measures the impacts of the respective alternative assets against the

standardised benchmark; rather than merely identifying whether Bitcoin enhances portfolio

performance or not. The following table summarises the data sourced from previous studies

evaluating the effectiveness of Bitcoin as a portfolio enhancer.

Table 1 – Previous Study Datasets

Brière et al. (2013) Chowdhury (2014) Wu and Pandey (2014)

Data Range July 2010 to July 2013 Aug 2010 to Jan 2014 July 2010 to Dec 2013

Stocks Stocks (Dev), Stocks (Emg) SP&500 S&P500

Bonds

Gov. B (Dev), Gov B (Emg), Inflation

Linked - World, Corp. B Global Bonds Global Bonds

Currencies EURO, JPY EURO, GBP USD w/ 10 currencies

Alternative Assets Gold, Oil, Real Estate, Hedge Funds

Gold Futures, Oil

Futures, Real Estate

Real Estate, S&P Volatility

Index (Fear Index)

Other Bitcoin Bitcoin Bitcoin

Table 1: The above table presents the data-sets used by previous empirical studies regarding Bitcoin within

a portfolio setting and as a diversifying tool. Specifically, the table displays the time range of the data sets

of the studies and the types of assets that were held in the test portfolios.

26

3.3. Benchmark Portfolio (K)

The data used is split into two groupings. The first set of data is sourced to emulate a

conservative UK investor portfolio – or benchmark portfolio (K). The constituent traditional

assets within this portfolio were selected due to the works of Ciner et al. (2013) and Marion

(2010). The resultant UK-centric investment thus mirroring a pension or hedge fund, the

constituents are:

Stocks:

FTSE All-Share Index

Bonds:

UK-GILTS – Risk-free

UK 10-Year Gov. Bond Index

Currency:

GBP/USD

Real Estate:

FTSE World Real Estate

The above benchmark portfolio is constructed in this fashion to emulate a traditional UK-

based hedge fund, pension fund or institutional investor with a low risk appetite. As

highlighted in the literature Evans and Archer (1968) demonstrate that portfolios with 10

constituents can effectively reduce levels of unsystematic risk to the point where only

systematic (or market risk) is present. As this empirical analysis is tailored to observe the

diversification benefits of alternative assets, it is appropriate to have some residual

unsystematic risk in order for the results to be both observable and significant. The number

of constituents of the benchmark portfolio is therefore set to 5. Constructing the portfolio in

this fashion gives a far more realistic portfolio than the portfolios other studies have used.

Emerging market stock indices and hedge funds for example are excluded from the

benchmark portfolio, due to the assumptions made about the specific investor appetite when

developing the benchmark portfolio.

Further justification for constructing the portfolio in this fashion is consistent with literature

sources highlighted in previous chapters. For instance, investors have a high tendency to

invest in funds and indices which they are highly familiar with (this is also referred to as the

home or familiarity bias). The portfolio categories chosen are also deemed to be the most

conventional and popular allocations for a UK investor – therefore this portfolio can be

deemed the UK benchmark of benchmark portfolios.

27

3.3.1. Descriptive Statistics - Benchmark Portfolio (K)

The undernoted descriptive statistics table shows the characteristics of the individual

constituents of the benchmark (K) portfolio. The Real Estate index has the highest return

(mean), yet it has the highest risk (standard deviation) and kurtosis. Unsurprisingly, the UK

Gilts – usually deemed to be risk-free benchmark – has the lowest risk, while matching the

FTSE All-Share in terms of return.

Table 2 – Descriptive Statistics – Benchmark Portfolio (K)

3.4. Alternative Assets - Test Assets (N)

The second set, the test assets (N), were chosen specifically with respect to their underlying

risks including volatility and unfamiliarity to a conservative investor. The test assets also

have widespread portfolio allocation within prior academic research and the aforementioned

literature relating to portfolio management. Essentially, these test assets will be evaluated

against the benchmark portfolio and generally consist of conventional alternative assets such

as - commodities, international bonds and currencies, as well as the inclusion of the

inherently risky and non-traditional wild-card investment - Bitcoin. These test assets will be

evaluated in the order shown. In my analysis, each test asset is treated individually in order

to keep statistics independent and unbiased. Specifically, the test assets are as follows:

Table 2: The above table presents the descriptive statistics of the portfolio holdings that compose the

benchmark portfolio (K) over the period July 2010 to January 2015.

FTSE All-Share UK Gilts UK 10-Yr Gov BI USD FTSE WLD RE

Mean 0.14% 0.14% 0.10% -0.01% 0.23%

Standard Error 0.0013 0.0005 0.0006 0.0006 0.0016

Median 0.0028 0.0016 0.0013 0.0001 0.0039

Standard Deviation 2.06% 0.78% 0.90% 0.98% 2.48%

Sample Variance 0.04% 0.01% 0.01% 0.01% 0.06%

Kurtosis 6.1734 0.6488 1.519 0.0249 9.0584

Skewness -0.8958 -0.3658 -0.5637 -0.1356 -0.7237

Range 0.1845 0.0467 0.0607 0.0546 0.2768

Minimum -0.1241 -0.0269 -0.0392 -0.0273 -0.1566

Maximum 0.0604 0.0198 0.0215 0.0273 0.1202

No. of Observations 235 235 235 235 235

28

Commodities:

Gold

Brent Oil

Currencies:

Euro/GBP

Japanese Yen/GBP

Developing Market Bonds:

Emerging Markets Bond Index

(EMG BI)

Alternative Asset Class

Bitcoin (BTC)

The economic benefits of incorporating these test assets within the aforementioned

benchmark portfolio (K) are evaluated - each test asset is considered in the prescribed order

above. This will be achieved by conducting the step-down mean-variance spanning method,

which is applied to test for spanning when investor preference follows a standardised utility

function.

As the above data was sourced in index format, it was appropriate to calculate the returns

using the following formula:

R = (Rt – Rt-1)/Rt-1) (1)

29

3.4.1. Descriptive Statistics - Test Assets (N)

Evaluating the descriptive statistics, it is evident that Bitcoin has extreme levels of both

return and risk in comparison to the other alternative assets. The asset with the lowest

standard deviation is the Euro, closely followed by the Emerging Government Bond Index.

Due to the negative average return of Brent Oil, it can be anticipated that it may perform

poorly in the following empirical analysis.

Table 3 – Descriptive Statistics – Test Assets (N)

As can be seen from the descriptive statistics, Bitcoin has the highest level of skewness.

Brière et al. (2010) explains that these high levels of skewness can only be achieved by

advanced strategies such as investments tuned to market volatility, which is in turn designed

to hedge portfolios against economic crises. In essence, this indicates that Bitcoin has the

potential characteristics to act as a possible safe-haven (or flight to quality) much like how

gold has been leveraged in times of economic instability. Moreover, Bitcoin has the highest

Sharpe ratio, which again is also a highly attractive consideration for investors.

Although Bitcoin has shown the highest levels of annualised risk in terms of standard

deviation of returns, it has substantially reduced over recent times, currently standing at

155%. Previous studies such as Chowdhury (2014), evaluating data from August 2010 to

January 2014, recorded some 258% volatility thus demonstrating the extent to which Bitcoin

has calmed in recent months. Further comparisons with this study reveal that kurtosis values

were at 16.10 during the worst of the price volatility and skewness levels during July 2013

stood around 2.30 for a significant period.

Gold Brent Crude Oil Euro Japansese Yen Emg Gov. Bonds Bitcoin

Mean 0.05% -0.16% 0.05% 0.13% 0.11% 5.50%

Standard Error 0.0016 0.0021 0.0006 0.001 0.0007 0.014

Median 0.0018 0.0007 0.0004 0.0002 0.0023 0.0123

Standard Dev. 0.0248 0.0322 0.0097 0.0147 0.0115 0.2150

Sample Variance 0.0006 0.001 0.0001 0.0002 0.0001 0.0462

Kurtosis 4.454 1.4893 0.7449 1.3164 9.6766 8.2715

Skewness -1.0385 -0.7833 0.2038 0.4154 -1.2945 2.1578

Range 0.2106 0.194 0.0687 0.1056 0.1214 1.6737

Minimum -0.1344 -0.1106 -0.0292 -0.0419 -0.0751 -0.5606

Maximum 0.0762 0.0834 0.0395 0.0637 0.0463 1.1132

No. of Observations 235 235 235 235 235 235

Table 3: The above table presents the descriptive statistics of the portfolio holdings that compose the test

assets (N) over the period July 2010 to January 2015.

30

3.5. Description of Correlation Matrices

Reviewing the correlation matrix of the benchmark and test assets combined, based on

weekly data from July 2010 to January 2015 (see Table 13 in Appendices), a number of

deductions can be made.

Firstly, it is clear that Bitcoin is highly dissociated with all other benchmark and test assets;

the closest correlation being with the FTSE World Real Estate Index (FTSE Wld. REIT)

0.1083 closely followed by Gold at 0.1054. This disassociation with other indices and assets

is where the value and effectiveness of Bitcoin as a diversifying tool comes from.

As the benchmark portfolio consists of predominantly UK-based assets, the portfolio is quite

significantly correlated. However, as confirmed by previous literature by Ciner et al. (2013),

the UK-denominated bonds demonstrate a negative correlation to the FTSE All-Share equity

index. This relationship is also to a lesser extent evident between Gold and the FTSE All-

Share, again verifying empirical findings from Erb and Harvey (2013) and Bauer et al.

(2010) - therefore creating an effective partial hedge against excessive volatility.

The close relationship between the Emerging Market Bonds with the rest of the assets is an

unexpected development and thusly may hinder its overall diversification benefits – this may

counter the empirical findings from Bekaert and Urias (1996). In addition to this, Real Estate

seems to be significantly linked to most assets and negatively related to bond prices.

2013 – 2014

During the period 2013 to 2014, the correlations (see Table 14 in Appendices) seem to

slightly increase between Bitcoin and UK Gilts and Gold at 0.2296 and 0.2536 respectively;

although overall this is still quite an insignificant relationship. Most notable however is the

relationship between the JPY and Bitcoin over this period at 0.3735, which is of great

significance in comparison to 0.0599 over 2010 to 2015. Effectively, this slight correlation

may indicate that the JPY and Bitcoin may behave similarly in the empirical tests for this

time period.

2014 – 2015

The correlations vary substantially over 2014 to 2015 (see Table 16 in Appendices) where all

the benchmark assets and test assets - besides UK Gilts, UK 10-Yr Gov. Bonds and Brent

Oil – have a distinctly negative relationship to Bitcoin. This signals that the results

concerning Bitcoin may significantly differ from the other tests assets.

31

3.6. Methodology

In order to evaluate the hypothetical benefits of investing in Bitcoin among other alternative

assets, I will undertake the step-down mean-variance spanning methodology proposed by

Kan and Zhou (2012). In essence, this is a statistical method that is used to test the impact

using historic data, of adding new assets into an existing portfolio of assets – as if the new

asset had been a component of the portfolio during the historic period under consideration.

As the Kan and Zhou (2012) test lends its methodology from the original test from

Huberman and Kandel (1987) it is appropriate to first review this approach and then develop

the discussion further to introduce the methodology undertaken for my research. Ultimately,

the Kan and Zhou (2012) test produces outputs which can be evaluated for significance in

reference to other test assets and the benchmark portfolio itself.

3.6.1. Huberman and Kandel (1987) Methodology and Formula

Taking the aforementioned into consideration, the formula as originally proposed by

Huberman and Kandel (1987) is defined by Kan and Zhou (2012) in the following fashion:

The returns of the benchmark portfolio (K) is denoted by a K x 1 vector signified by R2t,

while the test assets (N) are denoted by an N x 1 vector called R1t.

Hence the expected returns of N + K are denoted as follows:

μ = E[Rt] = [μ1 μ2] (2)

Therefore the respective covariance matrix of the combined N + K matrix is represented by

(where V is presumed to be non-singular):

V = Var [Rt] = (3)

We can present the following linear regression model when the benchmark and test assets

are combined accordingly:

R2t = α + β(R1t) + εt (4)

Essentially, this formula undertakes the assumption that E[εt] = 0N and E[εt R’1t] = 0NxK,

where 0 is an N-vector of zeros and 0NxK is an N by K matrix of zeros. In this context, the

error terms follow the normal distribution, are independent (uncorrelated) and

32

homoscedasticity exists. Also, let α and β to equal α = μ2 - β μ1 and β = V21V-1

11. Re-

arranging the regression formula alpha can be expressed as:

α = R2 - β(R1t) (5)

We can therefore infer that the spanning hypothesis occurs when the N alphas equate to zero

and the total sums of the beta co-efficient are equal to 1 for every asset. In order to

formulate the hypothesis, the delta term is set to δ = 1N - β1K, where 1N is an N-

denominated vector of ones.

Huberman and Kandel (1987) provide the appropriate conditions for spanning in terms of

restrictions on α and δ, where α signifies the tangency portfolio and δ denotes the global

minimum-variance portfolio. The null hypothesis stands as follows:

H0: α = 0N; δ = 0N

Essentially, if neither ‘α’ or ‘δ’ deviate from zero, or do not increase to significant levels

when adding the test asset (N) to the benchmark portfolio (K), it can be confidently

concluded that spanning is present. If there is a significant deviation at a 1, 5 or 10% level

from the initial equilibrium point, the null hypothesis can be rejected. Therefore the

alternative hypothesis can be accepted, implying that spanning does not occur and the test

asset enhances the portfolio’s mean-variance composition.

One downfall of this approach is that due to the constituent factors that derive δ, i.e. does not

involve the μ factor, this results in the δ factor being calculated with far greater accuracy

than α. In other words, the Huberman and Kandel (1987) spanning test places more weight

on the δ factor. Taking this into consideration, it is evident that when there is a statistically

significant impact upon the global minimum-variance portfolio, this does not always imply

economic importance. Alternatively, Kan and Zhou (2012) express that a large statistical

effect upon the tangency portfolios can be of economic importance, yet this may be

inherently more difficult to identify from a statistical point of view. Further discussion of the

limitations of this method can be located at Appendix 1.

In order to remedy the issues within this methodology and the other previous mean-variance

spanning test approaches, it is therefore appropriate to consider the step-down approach. The

following section will outline the data used to conduct the Kan and Zhou (2012) step-down

method of mean-variance spanning.

33

3.6.2. Kan and Zhou Step-down Application

As addressed in the literature review, the proposed mean-variance spanning methodology by

Kan and Zhou (2012) provides the most comprehensive and informative results. As such, the

step-down approach can be represented as follows:

The tangency portfolio is represented by α = 0 and F1 test evaluates whether the test asset

(N) has a significant impact upon the benchmark portfolio (K):

F1 = (T-k-1)(1/U – 1) (6)

This test is conducted assuming that “U is the ratio of the unconstrained estimate of variance

by imposing only the constraint of α = 0. Under the null hypothesis, F1 has a central F-

distribution with 1 and T-K-1 degrees of freedom” (Switzer and Fan, 2007, pp.107), where T

is the number of observations and K is the number of benchmark assets. Therefore, if the F1

test is has statistical significance, the two tangency portfolios are significantly different, i.e.

the test asset(s) (N) significantly improve the tangency portfolio efficiency.

The global minimum-variance portfolio is represented by ∑βj = 1 and is conditional that α =

0 under the F2 tests, when variance is ∑.

F2 = (T – K)(1/U – 1) (7)

The difference from the previous F1 test conditions is that ‘U’ is the “ratio of the constrained

estimate of variance by imposing only the constraint of α = 0 and the constrained estimate of

variance by imposing both the constraints of α = 0 and ∑βj = 1. F2 has a central F-

distribution and is independent of F1” (Switzer and Fan, 2007, pp.107); where T is the

number of observations and K is the number of benchmark assets. If the F2 test is rejected,

this demonstrates that the two global minimum variance portfolios (global minimum-

variance) are statistically different, informing the investor that the test asset (N) improves the

global minimum-variance.

Whereby the null hypothesis states that ‘α’ is a vector of zeros, and the delta is equal to a

vector of zeros. Therefore, spanning occurs when the investor does not benefit from

incorporating the new set of N risky asset(s) into the benchmark portfolio of K assets. Hence,

the null hypothesis is rejected if the investor receives any benefit (significance) from adding

N risky assets into the K benchmark of assets. In other words, the alternative hypothesis is

accepted if alpha or delta deviates from zero.

34

Therefore the spanning hypothesis of α = 0N, is a test of whether the global minimum-

variance portfolio has zero weights in the test assets. If the alpha term α = 0N is rejected and

accept the alternative, but accept δ =0N, then the improvement in investing in the test asset is

at the tangency portfolio.

In practical terms, the p-value tends to be denoted as 1, 5 or 10%; in this empirical analysis,

the p-value is set to 5%. Therefore, if the resultant p-value of the tests is less than 5%,

significance is present and therefore the null hypothesis can be rejected. For instance, a low

p-value in one of the aforementioned tests does not infer a large improvement in a mean-

variance frontier. A high p-value does not demonstrate that incorporating a test asset within a

portfolio will benefit the investor. Hence, it is far more conclusive to interpret the test

individually, which in turn will lead to more robust and informed investment decisions.

Again, a low p-value suggests some evidence of statistical significance, yet it does not imply

economic significance. Kan and Zhou (2001) state that assets with substantial F2 statistics do

not definitively infer any economic significance, yet significance for tests concerning the

tangency portfolios (significant F1 tests) may lead to economic benefits for risk-averse

investors.

To reiterate, since an asset may have an insignificant spanning result, coupled with a

significant F1 test, the step-down test is an enhanced approach to identify possible

diversification assets. Moreover, the step-down test also demonstrates a certain level of

sensitivity to the composition of the benchmark portfolios as well as highlighting acuteness

to trends over time. Thus separating the spanning test into two individual hypotheses by

using the F1 and F2 tests can lead to superior decision-making with regards to diversification

and portfolio composition.

If the initial hypothesis is rejected it can be inferred that the portfolio has increased expected

return for a similar level of risk. If the hypothesis at the global minimum-variance portfolio

is rejected, then it can be inferred that a portfolio can be devised with reduced risk for the

same level of expected return. Taking the aforementioned into account, the null and

alternative hypotheses can be defined as:

H0: Including alternative assets (and Bitcoin) within a conservative UK portfolio does not

increase mean-variance efficiency and Sharpe ratios.

H1: Including alternative assets (and Bitcoin) within a conservative UK portfolio increases

mean-variance efficiency and Sharpe ratios.

35

4. Empirical Results and Discussion

4.1. Introduction

The F1 test establishes whether there is statistical significance between the two tangency

portfolios K and K+N; whereas the F2 test highlights whether the two global minimum

variance portfolios are statistically different. When interpreting the Kan and Zhou (2012)

stepdown approach results, it is appropriate to define the level of significance for the F1 and

F2 tests. For this empirical analysis, the traditional significance level of 5% will be adopted

for both the F1 and F2 tests (similar to Kan and Zhou (2012)), although this significance

level can be appropriately set within the range of 1% to 10% depending on the preferred

level of confidence.

Another noteworthy aspect to consider when interpreting the results is the significance level

for the associative p-value for each of the F-tests. Therefore the p-value significance level

will be set to 5% also. In practice, this means that there is significance if the p-value is lower

than 5%. For the following results, one asterisk denotes significance to a 5% and two

signifies significance to a 10% level. In effect, these p-values help verify and ensure the F

tests are robust and to help mitigate the risk of accidently rejecting the null. As highlighted in

the aforementioned literature and the methodology, when both the F-test and the p-value

statistics are significant, the null hypothesis can be rejected; thus implying that adding the

test asset into the benchmark portfolio increases mean-variance efficiency.

For simplicity of interpretation, each test asset will be reviewed and critiqued in turn,

initially from the total time-series. In order to further inform deductions about each of these

test assets, and the factors that may have driven the results, sub-sets of each of the test assets

will also evaluated against their respective benchmark portfolios in the same empirical

fashion. Specifically, test assets were evaluated with respect to the two sub-sets of 2013 to

2014 and 2014 to 2015 in order to capture the period of Bitcoin’s price growth, decline and

price maturity and observe how this may impact portfolio allocation decision-making and

conclusions. These two periods capture the most turbulent and polar periods of Bitcoin’s

short lifespan; evaluating each sub-period separately has unveiled a number of interesting

results from investment and academic perspectives.

Having this supplementary information can enhance empirical understanding of Bitcoin’s

diversification suitability over time against other alternative assets. However, these data sub-

sets only consist of 52 observations each, which from an empirical perspective is lacking in

36

sufficient depth to reach solid deductions. For this reason, the sub-sets are lightly touched

upon to further aid the interpretation of the wider data-sets utilised in this study.

4.2. Gold

The first test asset evaluated – gold – has a highly significant F2 test and associative p-value,

i.e. the overall risk of the global minimum variance portfolio is reduced while maintaining

the same level of expected return. However, when comparing the benchmark (K) against the

benchmark and gold composed (K + Gold) efficient frontier, it seems to have made little to

no significant impact upon reducing the risk of the portfolio. Although the F2 test figure

demonstrates that gold has some potential to reduce risk when the expected return is low,

this should not affect investor decision-making as this would ultimately be sub-optimal. This

leads to questions on whether the Kan and Zhou (2012) methodology is statistically sensitive

in certain respects (see Table 11).

Furthermore, evaluating the data sub-sets over 2013 to 2015 (see Figure 11&17) established

also that gold has little impact at a 5% significance level. However, the p-values for 2013

and 2014 are significant at a 10% level which suggests that gold is marginally effective at

reducing risk in the benchmark portfolio. When evaluating the efficient frontiers for these

periods, however, there does not seem to be any notable improvements in mean-variance

efficiency.

Table 4 – F-test Results - Gold

Table 4: The associated table presents the F-test statistics from the Kan and Zhou (2012) step–down

approach described in Chapter 3 when the test asset (N) – gold – is incorporated into the benchmark

portfolio (K). One asterisk highlights significance at a 5% level and two asterisks denotes

significance at 10%. Note all p-values are exact under the normality assumption of the residuals. If the

F1 and associative p-value are significant to 5%, the tangency portfolio can be improved. If the F2 test

is rejected at a 5%, the global mean-variance can be improved.

Gold

F1 p-value F2 p-value

All 0.711 0.3999 14.398 0.000189*

2014 - 2015 1.539 0.2210 3.2610 0.0774**

2013 - 2014 3.232 0.0788** 2.898 0.0953

37

Possible reasons for this poor performance may derive from the fact that financial markets

have stabilised since the Global Financial Crisis of 2007-9, therefore implying that investors

are less reliant on gold as a safe haven, therefore decreasing demand and price accordingly.

Effectively, this confirms the findings from Bauer et al. (2010) who notes that gold has the

greatest demand in times of economic instability, increased demand partly driven by banks

reducing credit facilities, causing a shift to gold as a store of value. This poor portfolio

performance may have implications for how gold is treated in the future, especially as gold is

normally a consistent portfolio allocation for individual and institutional investors. This

confirms Caldwell’s (2015) view that gold may continue to fall, yet other experts believe that

gold has ‘bottomed out’ signalling a shift in momentum in gold price. Investors should

therefore closely monitor gold as a portfolio allocation going forward.

4.3. Brent Oil

The consistent drop in oil price over the past year driven by excessive supply and political

sanctions has placed it as the worst performing test asset in this empirical analysis.

Table 5 – F-test Results – Brent Oil

The insignificance of both the F1 and F2 tests and associative p-values confirm that spanning

is present when oil is incorporated into the benchmark portfolio and therefore demonstrates

no added improvement in mean-variance efficiency at this point in time, thus contradicting

the empirical findings from Arouri and Nguyen (2010). Reasons for these contrasting results

derive from the fact that oil prices reviewed in their study range from 1998 to 2007 (pre-

GFC), a period in which oil prices demonstrated significant price growth.

Table 5: The associated table presents the F-test statistics from the Kan and Zhou (2012) step–down

approach described in Chapter 3 when the test asset (N) – Brent Oil – is incorporated into the

benchmark portfolio (K). One asterisk highlights significance at a 5% level and two asterisks denotes

significance at 10%. Note all p-values are exact under the normality assumption of the residuals. If the

F1 and associative p-value are significant to 5%, the tangency portfolio can be improved. If the F2 test

is rejected at a 5%, the global mean-variance can be improved.

Brent Oil

F1 p-value F2 p-value

All 0.926 0.3368 0.083 0.7731

2014 - 2015 1.030 0.3155 1.5817 0.2147

2013 - 2014 1.061 0.3084 0.071 0.7910

38

Chart 3 – Brent Oil Market Price

The empirical results do, however, agree with the findings of Jones and Kaul (1996) who

claim that correlations between equity markets and oil prices fundamentally limit the

diversification benefits as a whole. It can therefore be inferred that investors should avoid

oil as a portfolio allocation for the time being, especially as oil prices continue to tumble.

The empirical results demonstrate that during the test period of 2010 to 2015 the

commodities – Gold and Brent Oil – performed worse than expected. Empirical findings

from Chen et al. (2002) and Fabozzi et al. (2008), among other scholars, indicate that

commodities in general significantly enhance the expected return and reduce the risk of risky

portfolios – clearly this is not the case from the empirical results in this study. Reasons for

these differences may derive from the prevailing market conditions as well as other

economic factors. Thind (2014) highlights that due to the poor performance of the global oil

market, wealth fund managers have been forced to re-examine their portfolios and have

turned to towards alternative investments to hedge their risk exposures.

However, not all literature conveys results that fully agree with the above. The recent

empirical findings from Cheug and Min (2010) details that commodities demonstrate poor

diversification benefits in the short-term, yet tend to perform much more effectively over the

longer-term. Therefore it can be implied that commodities should still be considered as a

portfolio addition, albeit a passive portfolio geared towards long-term gains.

Chart 3: The above chart presents the market price of Brent Crude Oil Spot price developments in

U.S. dollars over the period 1998 to 2015 (Brent Crude Oil Spot, 2015).

39

4.4. Euro

Overall, the Euro showed a significant F2 test and p-value statistics thus implying an

enhancement at the global minimum-variance portfolio. Further evidence from the 2013 to

2014 sub-set reveals that it was the only statistically significant F2 Test for this time period,

standing at 15.837. Therefore it can be inferred that the Euro is highly effective at reducing

portfolio risk, therefore the null hypothesis can be rejected with confidence.

Table 6 – F-test Results - Euro

However, once more there seems to be contradictory conclusions drawn from alternative

approaches to identifying mean-variance efficiency. For instance, in the 2014 to 2015 sub-

set, the F1 and F2 statistics confirm that the Euro lacked sufficient impact upon the

benchmark at a 5% significance level; yet, reviewing the efficient frontier (see Figure 19)

signalled indications of a reduction in risk. These results establish that deciding upon the

level of statistical significance is of high importance, as it may lead to the false rejection or

acceptance of the null hypothesis. To mitigate the issue of misinterpretation, it is essential to

conduct as many mean-variance efficiency tests and performance measures as possible.

4.5. Japanese Yen

One of the most consistent mean-variance efficient test assets throughout the empirical

analyses was the Japanese Yen (JPY). Not only was it a strong performer, the JPY was

exclusively the only test asset to enhance both the expected return of the portfolio while

maintaining the same level of risk (tangency) and reducing risk while maintaining the same

expected return (global minimum-variance portfolio).

Euro

F1 p-value F2 p-value

All 2.388 0.1237 31.741 5.124E-08*

2014 - 2015 2.427 0.1261 0.6615 0.4201

2013 - 2014 0.499 0.4834 15.837 0.000238*

Table 6: The associated table presents the F-test statistics from the Kan and Zhou (2012) step–down

approach described in Chapter 3 when the test asset (N) – Euro – is incorporated into the benchmark

portfolio (K). One asterisk highlights significance at a 5% level and two asterisks denotes significance at

10%. Note all p-values are exact under the normality assumption of the residuals. If the F1 and associative

p-value are significant to 5%, the tangency portfolio can be improved. If the F2 test is rejected at a 5%, the

global mean-variance can be improved.

40

Table 7 – F-test Results – Japanese Yen

For instance, evaluating the 2014 to 2015 sub-set illustrates that the JPY significantly

improved the benchmark portfolio at the tangency (8.367) portfolio and was of borderline

significance at the global minimum-variance (4.38) portfolio. These results mirror the

academic coverage highlighted in the literature, that the JPY is a reliable safe-haven

component to investment portfolios due to its disassociated relationship and unique

behaviour with respect to other global currencies.

Although the majority of previous mean-variance spanning results provide highly logical and

consistent statistics, which to an extent, match the conclusions with the corresponding

efficient frontiers (see Figures 6, 14 and 20). The 2013-2014 sub-set concerning the JPY is a

significant outlier. Specifically, the F-test statistics indicate that the JPY was largely

insignificant, while the efficient frontier highlighted that JPY was a highly effective

allocation – reducing both the risk and increasing the expected return. Although this

dilemma has been discussed with previous assets, the scale of the misrepresentation is highly

irregular. It could therefore be inferred that the Kan and Zhou (2012) step-down approach

has an underlying methodological or mathematical bias which leads to systematic errors in a

number of cases. However, due to the complexity of the methods undertaken by Kan and

Zhou (2012), it is inherently difficult to accurately pinpoint the source of mathematical or

systematic error.

The Euro and the Japanese Yen were both highly effective at diversifying the benchmark (K)

portfolio at the global-minimum variance portfolio. This finding agrees with the empirical

evidence presented by Marion (2010) and Stubbington (2014) which signals that both the

Euro and the Japanese Yen are deemed safe havens during times of economic stability and

volatility.

JPY

F1 p-value F2 p-value

All 5.347 0.0216* 18.059 0.000031*

2014 - 2015 8.367 0.00582** 4.3827 0.0417*

2013 - 2014 0.726 0.3988 0.879 0.3533

Table 7: The associated table presents the F-test statistics from the Kan and Zhou (2012) step–down

approach described in Chapter 3 when the test asset (N) – Japanese Yen (JPY) – is incorporated into the

benchmark portfolio (K). One asterisk highlights significance at a 5% level and two asterisks denotes

significance at 10%. Note all p-values are exact under the normality assumption of the residuals. If the F1

and associative p-value are significant to 5%, the tangency portfolio can be improved. If the F2 test is

rejected at a 5%, the global mean-variance can be improved.

41

4.6. Emerging Market Bond Index (EMGBi)

It is observed that the EMGBi allocation, as a whole, was a poor portfolio inclusion with no

statistical significance within the spanning tests; although eyeballing the efficient frontier

(see Figure 7) deviation, would suggest an indication of efficiency improvement to some

degree.

Table 8 – F-test Results - EMGBi

It should therefore be noted that the EMGBi would have been deemed significant if a higher

significance value was selected e.g. 10%. Although the F1 and F2 tests were held to show

minimal significance at 5%, a level of 3% would have resulted in a rejection of the null

hypothesis. Evidently, the significance level selected is dependent upon investors’

preferences and risk-appetite; whereby a risk-seeking investor will desire a significant F1

test and a risk-averse investor appeals for a significant F2 statistic. By reducing the

significance level from 5% to 1%, it will result in the rejection of the null hypothesis far

more frequently, therefore exposing the investor to more risky asset options. It can therefore

be inferred that a risk-averse investor will be less susceptible to change the statistical

significance of the mean-variance tests to increase confidence that the test assets they

undertake can reduce the risk of their portfolios. However, as mentioned earlier, investors

should be aware of the statistical discrepancy which can occur with a number of outlying test

assets.

Essentially these findings fundamentally disagree with the empirical evidence presented by

Solnik (1974) and Driessen and Laeven (2007), which detail how portfolio performance can

be enhanced by having exposure to international markets. Although these studies document

the added benefits of diversifying with foreign market equity indices, these findings are also

expected to hold true with international and emerging market bond indices. Evidently, the

Table 8: The associated table presents the F-test statistics from the Kan and Zhou (2012) step–down

approach described in Chapter 3 when the test asset (N) – Emerging Market Bond Index (EMGBi) – is

incorporated into the benchmark portfolio (K). One asterisk highlights significance at a 5% level and

two asterisks denotes significance at 10%. Note all p-values are exact under the normality assumption

of the residuals. If the F1 and associative p-value are significant to 5%, the tangency portfolio can be

improved. If the F2 test is rejected at a 5%, the global mean-variance can be improved.

EMGBi

F1 p-value F2 p-value

All 0.304 0.5822 0.620 0.4317

2014 - 2015 1.873 0.1778 2.2979 0.1362

2013 - 2014 1.582 0.2148 3.268 0.077**

42

correlation between markets, as proposed by Christofferson et al. (2010), has increased to the

extent where there are insignificant diversification benefits for a UK investor to have

exposure to emerging market bonds.

4.7. Bitcoin

Out of all of the assets evaluated, the greatest performer was Bitcoin by a significant margin.

The F1 test statistic for Bitcoin was the highest out of all of the empirical analyses, thus

demonstrating that the qualities of Bitcoin make it highly effective at enhancing both the

tangency and global-minimum variance portfolios.

Table 9 – F-test Results - Bitcoin

Bitcoin (2013 to 2014) – During this period, Bitcoin saw mass popularity among mainstream

news services, pushing the price up to levels around $1,000 in just under a 2 month period.

In the empirical analysis, Bitcoin has highly significant F1 statistics highlighting that during

this period investors would have greatly benefitted at the tangency portfolios (see Figure 16).

Bitcoin (2014 to 2015) – During this time-period, Bitcoin price has been on a downward

trajectory. This has undoubtedly stunted its ability to improve portfolio performance.

Specifically, Bitcoin during this time period demonstrated the lowest statistical significance

for both the F1 and F2 test (see Figure 22).

As a stand-alone review, an optimal portfolio was devised which consisted of the benchmark

portfolio and all of the test assets. The optimal portfolio with these constituents detailed that

an allocation of 3% invested in Bitcoin resulted in the highest Sharpe ratio, thus verifying the

results of Brière et al. (2013). On the whole, the extensive diversification benefits from

Bitcoin

F1 p-value F2 p-value

All 12.105 0.0006* 2.085 0.1501

2014 - 2015 0.300 0.5862 0.0357 0.8508

2013 - 2014 6.973 0.0113* 3.646 0.0623**

Table 9: The associated table presents the F-test statistics from the Kan and Zhou (2012) step–down

approach described in Chapter 3 when the test asset (N) – Bitcoin – is incorporated into the benchmark

portfolio (K). One asterisk highlights significance at a 5% level and two asterisks denotes significance at

10%. Note all p-values are exact under the normality assumption of the residuals. If the F1 and associative

p-value are significant to 5%, the tangency portfolio can be improved. If the F2 test is rejected at a 5%, the

global mean-variance can be improved.

43

Bitcoin therefore confidently agree with the findings from Brière et al. (2013), albeit from an

alternative and more comprehensive empirical perspective. The above empirical results also

indicate that during the time-period evaluated, Bitcoin outperforms all the alternative assets,

an area which Brière et al. (2013) left inconclusive in the published work.

Having taken all of the aforementioned into consideration, it is evident that Bitcoin

significantly improves the mean-variance efficiency of the UK investor’s portfolio and this

means that it is worthy of consideration by all types of investors going forward. Most

notably, Bitcoin can be highly effective for an investor seeking to improve their tangency

portfolio, yet caution should be taken if the investor wishes to improve their global

minimum-variance portfolio.

As aforementioned by Kat (2006) investors with limited knowledge tend to review

alternative assets in a similar fashion to government bonds and equities. Therefore the above

empirical findings do not indicate that Bitcoin will immediately find favour with investors

who have a conventional approach to evaluating potential investments. Taking all of the

aforementioned into account it is evident that investors can reap significant diversification

benefits, especially when an investor slightly increases their risk tolerance, they can realise

significant benefits to their portfolios.

4.8. All Test Assets in the Benchmark Portfolio

When all 6 test assets were incorporated within the benchmark (K) portfolio, mean-variance

efficiency was significantly improved at both the tangency and global-minimum variance

portfolios (see Figure 1).

Table 10 – F-test Results – All Test Assets (N)

Table 10: The associated table presents the F-test statistics from the Kan and Zhou (2012) step–down

approach described in Chapter 3 when all test assets (N) – Gold, Brent Oil, the Euro, the Japanese

Yen, EMGBi and Bitcoin – are incorporated into the benchmark portfolio (K) over the period 2010 to

2015. One asterisk highlights significance at a 5% level and two asterisks denotes significance at

10%. Note all p-values are exact under the normality assumption of the residuals. If the F1 and

associative p-value are significant to 5%, the tangency portfolio can be improved. If the F2 test is

rejected at a 5%, the global mean-variance can be improved.

All Test Assets

F1 p-value F2 p-value

2010 to 2015 20.0295 0.000012* 55.621 1.783E-12*

44

Specifically, the resultant F1 test figure and associated p-value had by far the highest

expected return increase relative to similar risk levels. This result was also evident with the

F2 which finalised at 55.621 with a respective p-value of 1.783E-12 which again stands far

higher than all the individual empirical tests conducted. Therefore for both instances the null

hypothesis (that spanning is occurring) can be rejected and the alternative hypothesis

accepted.

However, as highlighted previously by De Roon et al. (2001) and Kan and Zhou (2012), as

the number of test assets increases above n=1, the likelihood of estimation errors increasing

is compounded, thus resulting in co-efficient values and regressions which may be prone to

systematic errors. Effectively, this may limit the reliability of interpretation and drawing

conclusions from this specific test which incorporates 6 test assets in isolation.

Chart 4 – Optimal Portfolio Weightings

To mitigate this issue and to observe the constituent assets of the optimal portfolio, an

efficient frontier was constructed. As can be seen the optimal risky portfolio (tangent) is

predominantly composed of a high allocation of low-risk UK Gilts as well as the FTSE All-

Share and Japanese Yen. Among these is a small weighting in Bitcoin, thus confirming that a

conservative portfolio can reap significant benefits without overexposing to risky strategies.

Chart 4: The associative chart presents the weights of the optimal portfolio when all test assets (N) –

Gold, Brent Oil, Euro, Japanese Yen, EMGBi and Bitcoin – are incorporated within the benchmark

portfolio (K) over the period 2010 to 2015.

45

4.8.1. Short Selling

There is significant improvement in portfolio performance when the short selling constraint

assumption is relaxed. From intuition, this should be expected as shorting effectively allows

an investor to profit both when asset prices are rising and falling. Most interestingly, going

short on Bitcoins at a level of around 1% helped significantly reduce the standard deviation

(risk) at low return levels. On average the portfolio which permitted short trading was 6.77%

more effective at reducing risk. From eyeballing the efficient frontier (see Figure 10) with

the relaxed short-selling constraints, it is evident that short selling requires alertness to

market developments, as the efficient frontier demonstrates a level of jitteriness and

irregularities when related to the uniformity of the long-only portfolio. These findings are

also mirrored when all test assets (N) were subject to these same test conditions (see Figure

2).

As it has been assumed that the UK-centric investor is risk-averse and lacking in financial

skills, it can be inferred that shorting may be beyond the competency of this type of investor.

However, for those more adept to financial analytics and practice - namely institutional

investors – it is clear that Bitcoin has the potential to significantly improve and reshape the

direction of composing both risky and low-risk portfolios. Taking this into consideration, it

can be conjectured that Bitcoin may not have the most suitable characteristics for a buy-and-

hold portfolio, yet for an actively-managed portfolio it may be highly effective.

However, outwith the hypothetical portfolio scenario short-selling is a far more complicated

avenue. Specifically, the highly complex nature of calculating optimal portfolios with short-

selling, output can vary significantly, resulting in different statistics and variables each time

a calculation is run, thus increasing the likelihood of estimation error. Moreover, in real

practice, there may be restrictions with respect to shorting. Not all assets are highly

transparent and liquid enough to permit short selling.

46

4.9. Limitations and Other Observations

On review, it is evident that there are a number of constraining factors that directly impact

the empirical results; these are discussed below.

First of all, the hypothetical UK conservative portfolio was constructed purely through using

prior knowledge and intuition surrounding the concept of a UK-based and UK-centric

conservative institutional investor. It is inherently difficult to accurately obtain the

constituent assets holdings of hedge and pension funds, as this information is not usually in

the public domain, being closely guarded industry secrets. Having a greater insight into

actual hedge funds would have undoubtedly resulted in more representative results and

would have helped bridge the gap between the empirical findings in this work and

investment returns earned in practice. It can also be argued that restricting the benchmark

portfolio to a UK perspective may also limit the value of this research. Extending the

benchmark portfolio and scope of research to incorporate other international market

perspectives, or even accommodate a globally-oriented investor, would have enhanced the

research further.

The restricted time-series used in this empirical analysis may lead to limited empirical

robustness – however, this is constrained by the limited lifespan of Bitcoin and the time-

series selected was therefore chosen to maximise the period for which Bitcoin data is

available. Extending the scope and variety of the test assets used may have given rise to

additional results with improved robustness.

Implementing the spanning tests through advanced software packages such as MatLab,

would have greatly increased my ability to conduct a broader range and variation of mean-

variance spanning tests, which would have given added weight and depth to the

interpretation of the empirical results. Some of these tests are the: Huberman and Kandel

(1987) approach, Wald Test, the Likelihood ratio and the Lagrange Multiplier, as highlighted

in the Appendix 1. However, due to the time constraints, this was beyond the scope of this

research.

Although the Kan and Zhou (2012) approach provides a revised and more informative

methodology than other mean-variance spanning approaches, it still has inherent limitations

in practice. Specifically, in a number of instances it was clear that the F-test figures derived

from conducting the Kan and Zhou (2012) step down approach and the associated plotted

efficient frontiers, conveyed differing and inconsistent results. These outstanding,

contradictory results in effect pose either an issue in decision-making on whether a particular

47

significance level is justified, or there are residual estimation errors within the mathematical

framework in the mean-variance spanning methodology. It is crucial not to accept the results

of the Kan and Zhou (2012) approach in isolation or accept the statistical outputs at face

value. To mitigate this issue, it is clear that the Kan and Zhou (2012) approach outcomes are

heavily dependent upon the chosen level of significance, which in turn can dictate the

resultant interpretation of the findings. It is therefore important to critically appraise multiple

variations of mean-variance spanning tests, among other informative statistical performance

measures. This will undoubtedly supplement the investor’s decision-making, help to justify

use of specific significance levels and better inform investment decisions.

At times it is evident that some alternative assets provide significant diversification benefits;

however, the results do not indicate or provide a measure of how large the benefits are in

reality. This leaving the reviewer in the dark as to how much he/she should allocate to

realise these benefits. Moreover, using historic data-sets may explain how a hypothetical

portfolio would have performed in the past with inclusion of the assets under test, yet, it is

unclear whether these recommendations will hold fast going forward. Essentially, it is easy

to form deductions in hindsight, yet predicting the future value and volatility of assets such

as Bitcoin is still highly uncertain in practice: the past potentially being an unreliable guide

to the future. However, investors do make extensive use of historic data and use advanced

charting techniques to look for short and long-term trends, resistance levels and support

levels. Investors also have extensive analytical resources available to perform studies on

whether or not to invest in particular assets and when to enter the market and also when to

exit a position as well as deciding to go short or long. Such investors will usually also have

available professional advice and excellent sources of data and news feeds with detailed

economic insight and forecasts. Taken together, the sum of all such resources should enable

investors to make sound and balanced decisions on the future direction of both their longer-

term strategies and short-term tactical manoeuvres in terms of portfolio asset holdings,

allocation and risk profile.

An important point to take into consideration is that Bitcoin’s success, i.e. increase in

demand which in turn drives its value, is largely correlated to technological, economic and

regulatory developments. These have been touched upon to some extent by previous

research (see Appendix 6), yet these aspects of Bitcoin are beyond the scope of this research

paper. Therefore it is essential for investors to have a broad and up-to-date understanding of

the fundamental dynamics of the cryptocurrency and Bitcoin markets in order to stay up-to-

date with market shifts, developments and opportunities which may occur at a rapid pace.

48

Cryptocurrencies themselves are likely to evolve rapidly, with new developments both

exploiting opportunities and also closing any problems or issues identified with Bitcoin as it

matures through the economic cycle.

4.10. Summary of Empirical Results

From reviewing the results obtained, it is evident that mean-variance spanning tests provide

an effective means to identify whether incorporating test assets within a benchmark universe

of assets are beneficial in terms of diversification and thereby lowering portfolio risk.

However, there are numerous issues surrounding how the end-user should interpret the final

results of these tests.

Gold, Brent Oil and the Emerging Market Bond Index all performed worse than expected in

terms of diversification benefits. Obvious reasons for this result from an economic

standpoint include falling demand for gold as a safe haven, as western economies continue

their recovery and the recent steep decline in oil prices caused by oversupply and falling

demand in key growth economies such as China and exploitation of alternative energy

sources such as shale gas in North America. The Emerging Market Bond Index was

expected to show better diversification benefits than it did in practice, the observed outcome

probably demonstrating globalisation effects and increasing correlation with western

economies.

In comparison, the Euro and the Japanese Yen performed substantially better than initially

thought, and were generally deemed effective at reducing portfolio risk at the global

minimum variance portfolio – thus confirming findings from Marion (2010), albeit from an

alternative empirical approach with different assets and portfolio constructs.

The most profound deduction made from the consideration of the empirical analysis is that

Bitcoin is a highly effective tool, both in reducing portfolio risk and enhancing the overall

return of the portfolios. These results confirm that Bitcoin has both the credibility and

potential to supplement an investor’s diversification options. In addition, Bitcoin also

excelled in the combined portfolio test, thus highlighting its inherent diversifying qualities

amongst other credible alternative assets. Predominantly, the combined portfolio consisted of

safer allocations such as UK Gilts and the Japanese Yen, while Bitcoin with a comparably

small allocation of 3.08%, proved that it is a worthy constituent within a portfolio, with good

positive benefits in terms of diversification and lowering risk.

49

Moreover, to supplement rounded understanding of how investors can actively manage their

investments in Bitcoin using other techniques available to them, it was also appropriate to

review how the portfolio performs when short selling restrictions are relaxed.

Unsurprisingly, both return and risk were further enhanced, thus confirming theory and

verifying that portfolio managers have opportunities to profit from Bitcoin in bear market

conditions by using short positions to their advantage.

The conclusions reached following analysis of portfolio diversification indicate that there are

advantages to be gained by utilising alternative assets-types. It will require a change in the

mind-set of investment professionals before there is substantial change in attitude and greater

use of alternative investment types and some of the reasons for such observed behaviours

and effective impediments to progress are also discussed in the concluding chapter.

50

5. Conclusions

5.1. Introduction and Summary of Research

The literature on the highly topical area of diversification underlines the importance of

managing the risk and returns inherent with investment portfolios. Recent academic findings

and the Global Financial Crisis of 2007-9 have emphasised that the global integration of

markets has progressively increased the correlations of assets, thus increasing the difficulty

in maintaining manageable levels of diversifiable risk. In order to investigate this issue, an

empirical comparison was conducted in light of alternative assets and the new

cryptocurrency - Bitcoin. Literature also highlighted that the most effective means of

evaluating portfolio performance is through conducting mean-variance spanning tests.

Specifically, The Kan and Zhou (2012) step-down approach provided the most empirically

robust means of evaluating the mean-variance relationship between two portfolios. Among

the test assets reviewed, Bitcoin, the Japanese Yen and the Euro demonstrated the greatest

significance and credibility in improving portfolio performance.

Taking the output of the research work into consideration, it is clear that all breeds of

investor have increasingly complex factors to weigh-up and interpret when considering their

approaches to risk in general and particularly when diversifying their portfolios – especially

as the financial landscape becomes more intertwined and confusing. Specifically, the ever-

increasing correlation of the global financial market, as presented by Bernstein and

Pinkernell (2007), and the speed of knowledge transfer are making the art of diversification a

far more elusive and difficult practice and therefore making success harder to achieve.

Observing the substantial growth of financial markets and ongoing innovation, it is clear that

financial practices will become ever-more competitive. Barriers to entry and transaction

costs will continue reducing, more and more practices will move to real-time and investor

appeal will widen in terms of the expanding universe of tradable asset types. In response,

investment practitioners need to become more adept at anticipating, identifying and

mitigating risk if they are to avoid constructing portfolios that are unwittingly exposed to

hidden dangers contributed to by increasing correlation between assets and international

markets – correlations that may be difficult to detect.

Ample academic findings and the accelerating nature of financial markets have indicated that

old conventions of investing and measuring risk do not necessarily stack-up in real world

scenarios. Thus it is necessary that the next generation of investors adopt a far more open

and lateral mind-set in regards to risky assets and portfolio diversification in general.

51

Evaluating the financial instruments an ordinary investor can undertake positions with today,

it is evident that there are exchanges where investors can take long and short positions in a

wide variety of assets including – commodities, global share indices, currency pairs and

company stocks. Investors can easily take short or long positions, invest in real-time in

either current or future values and use various instruments tuned to their personal strategy,

trading style and attitude to risk. As well as futures, today investors can utilise traded

instruments such as contracts for difference (CFDs) which are leveraged products and mean

that the investor can invest in a commodity, currency or index without any need to take

actual ownership of the underlying assets concerned; this allows easy entry and exit from

positions and many such CFDs have tight spreads which also limit the costs of investing,

further widening appeal. Research carried out as part of this work has also highlighted that

Bitcoin futures, CFDs, binaries and spread-betting options have now been introduced on

some on-line trading exchanges.

5.2. The Future Professional Investor

The next generation of investors and portfolio managers will have grown up with and will be

more comfortable with exposure to constant technological innovation, the real-time

availability of data and the 24/7 nature of global financial markets. This will allow them to

generally be in better positions to capitalise on opportunities as and when such opportunities

arise and may also mean that they have a natural leaning towards more innovative products

and digital concepts, because they have grown up in the digital ‘always-on’ age, whereas

older generations may have less of an appetite for such innovation. This development and

tendency is evident from the research carried out as part of this paper in terms of the

adoption of Bitcoin and its appeal in terms of demographics. The Generation X and baby-

boomer generations, who are now at an age and experience-level where they are more likely

to be in charge of global portfolios and investment decisions for multinational companies,

seem to have an awareness of Bitcoin but to date deem it too unreliable and too unstructured

to consider for incorporation into their portfolios. This is possibly owing to their tendency to

stick with what they know in what could be termed a linear mind-set.

Perhaps the most ground-breaking implication derived from the empirical analysis conducted

as part of this research is that essentially, future investors need to take on board some risks in

order to reap the diversification benefits they are seeking. This will therefore include using

and benefitting from risky assets such as Bitcoin – otherwise, investors will have difficulty in

52

outperforming traditional benchmark portfolios and the risk-free rate. This may require

either a shift in mind-sets of investors, or a prolonged pause until the current generation of

investment professionals are replaced by the millennial generation. In the shorter-term, more

of an attitudinal shift is required in terms of deliberately and consciously assuming some risk

within a portfolio. With such a strategy, there needs to be acceptance that it is allowable to

occasionally ‘get it wrong’ with asset choices, with occasional poor performers and loss-

making positions – such losses can be mitigated with market limit orders for pre-determined

stop-loss levels. Allocation and diversification reduce the downside risk impact of such

strategies and are actually a sign of getting the diversification strategy right – occasional

failures are to be expected as the ‘norm’ because without the occasional failure the strategy

is too conservative and risk-averse to reap optimal benefits.

Having empirically investigated new financial innovations and alternative assets, such as

Bitcoin, it is evident that both portfolio risk and return levels can be significantly enhanced.

These findings therefore agree with Anson (2006) who claims that investors that are willing

to investigate new alternative investments stand in a greater position to mitigate the risk of

growing levels of correlation between assets in the global economy as well as attaining

superior returns. Specifically, as the growing popularity of and credibility of

cryptocurrencies and other alternative assets increases, this may provide a new avenue for

investors to maintain and/or enhance optimal risk and return levels.

5.3. Diversification Benefits

This study has also provided supportive evidence to suggest that the Japanese Yen and the

Euro are commendable assets in terms of safe haven currencies (Marion, 2010). The reasons

for this result and performance in the period under review are essentially the conservative

approaches taken on the whole by the governments of Japan and most of the Eurozone

countries, although there will be a degree of variability within the Eurozone. Specifically,

Jones (2015) indicates that the main economies of Germany and France are large enough to

smooth out some of the peaks and troughs caused by the smaller weaker members.

Although the findings of this study based on the data-sets analysed strongly suggest Bitcoin

is a highly effective diversifying asset, this does not necessarily imply Bitcoin may continue

to be as effective and provide similar diversification benefits in the future in view of

inevitable differences in future economic conditions. Although from following Bitcoin’s

price development, it seems that Bitcoin has passed through and cleared the stormy periods

53

of acceptance and volatility caused by events such as governmental action and thefts of

Bitcoin from trading exchanges, it is of high probability that Bitcoin will be exposed to

further turbulence in years to come. It is also possible that the correlation with other

financial instruments will increase as Bitcoin transactions increase, it gains wider acceptance

among major retailers and appeals to a wider investor audience with corresponding

sentiment shifting towards traditional levels in terms of investor mind-set.

5.4. Bitcoin and Future Entrepreneurial Developments

Personal communications conducted as part of the research for this work, via Skype calls

with Bird (2014) - a Bitcoin enthusiast based in New Zealand - helped highlight that Bitcoin

initially was the preserve of computer software programmers and that the initial appeal has

now widened extending mainly to the technologically-minded within the millennial

generation. Also, this part of the research indicated that as an instrument, Bitcoin has the

potential to become a disruptive innovation, which may in turn reshape the financial

landscape. These findings are encouraging and tend to indicate that the platform established

by Bitcoin will be built upon both by Bitcoin itself and also by new products, markets and

developments in general, which at this time are unknown but will happen at pace spurred on

by the speed of modern communication and social networking.

5.5. Suggestions for Further Research

As Bitcoin’s price has shown promising signs of stabilisation in recent months, Wolman

(2015) suggests that this may signal that Bitcoin is entering the maturity stage of its

development cycle, with wider investor appeal and acceptance starting to show through in

terms of lower volatility and corresponding price stability. In effect, as Bitcoin’s price

volatility calms, news-feeds and general data availability about it settle down, its credibility

as an effective and efficient method of transferring economic value may increase further and

may even increase significantly. This in turn, will enhance its ability to diversify portfolios

without the excessive volatility experienced historically. There are also easier routes of entry

for investors, owing to its wider availability and increasing ability to invest in it as a

currency pair against say the US Dollar, Euro, British Pound or Japanese Yen or gain

exposure to Bitcoin in the secondary market (Primack, 2013). From an empirical standpoint,

having a less volatile data-set of price movements will ultimately improve the

54

representativeness of the findings and should allow Bitcoin’s inclusion on the agendas of

more investment committees.

Evaluating a time series beyond the 235 weekly observations used within my research should

improve the robustness of the mean-variance spanning tests conducted, thus increasing the

value of the deductions highlighted in the results and the applicability of the research

findings. Additionally, reviewing further alternative assets - as well as other cryptocurrencies

- within the empirical analysis would have also added further depth and value to this

research. However, at the date of writing insufficient data is available for other

cryptocurrencies to facilitate robust research and warrant inclusion.

Further research could be carried out to devise a mean-variance spanning approach that

provides, in addition to providing indications of suitability for diversification and risk

reduction, a quantitative indication as to how much an investor should invest into a test asset

to derive target benefits. This method, if coupled with an ability to accurately model the

investors’ risk appetite and risk tolerance, would give a powerful indicator that investors

could utilise as an input to their decision-making. Essentially, this would be one of a number

of inputs, but it would facilitate alternative investments such as Bitcoin getting on the agenda

at board meetings and in investment committees across the globe. Moreover, having

established results from both the U.S. and now the U.K. perspective, it would highly

enlightening to visualise how international markets can utilise and benefit from an allocation

of Bitcoins. Perhaps an even more ambitious research area could evaluate and compare the

performance of an international portfolio when incorporating Bitcoin.

This type of empirical analysis, to my knowledge, is the first research to evaluate Bitcoin in

this manner and should act as a foundation for further leap-frog studies to progress this niche

area of financial research further. It is clear that as the global marketplace becomes ever

more correlated the financial world will continue to turn to new alternative means of

achieving sufficient levels of diversification. This research has demonstrated that Bitcoin can

add to the portfolio manager’s arsenal of diversification options. Looking forward to the

future I believe that some revised form of cryptocurrency, inspired by the foundational

philosophy of Bitcoin and what it stands for, will eventually gain traction and thusly change

the way in which the financial world and associated portfolio management systems operate.

55

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7. Appendices

Page Number

Appendix 1

Other Methods of Mean-variance spanning 67

Appendix 2 Bitcoin – Technical

68

Appendix 3 Risks with Bitcoin

68

Appendix 4 Mining and Blockchain Technology

70

Appendix 5 Bitcoin Exchanges

71

Appendix 6

Cyber, Regulatory, Political and Ethical

Implications of Bitcoin

71

Appendix 7 Other Cryptocurrencies

72

Appendix 8 Behavioural Finance and Bitcoin 73

Glossary

75

Glossary 1

Glossary 2

Portfolio Theory - formulae

Risk free rate

75

76

Glossary 3 Sharpe Ratios 76

Tables

Table 11 Kan and Zhou (2012) Step-down Tests

77

Table 12

Constituents of Optimal Portfolios 78

Table 13 Correlation Co-efficient Matrix – 2010 to 2015

79

Table 14 Correlation Co-efficient Matrix – 2013 to 2014

80

Table 15 Descriptive Statistics – 2013 to 2014

81

Table 16 Correlation Co-efficient Matrix – 2014 to 2015

82

Table 17 Descriptive Statistics – 2014 to 2015

83

66

Figures – Efficient Frontiers (2010 to 2015)

Figure 1 Benchmark Assets (K) + All Test Assets (N) - (2010 to 2015)

84

Figure 2 Benchmark Assets (K) + All Test Assets (N) – (2010 to 2015)

[Short Sales constraint relaxed]

85

Figure 3 Benchmark Portfolio (K) + Gold (N) – (2010 to 2015)

86

Figure 4 Benchmark Portfolio (K) + Oil (N) – (2010 to 2015)

87

Figure 5 Benchmark Portfolio (K) + Euro (N) – (2010 to 2015)

88

Figure 6 Benchmark Portfolio (K) + JPY (N) - (2010 to 2015)

89

Figure 7 Benchmark Portfolio (K) + EMGBi (N) - (2010 to 2015)

90

Figure 8 Benchmark Portfolio (K) + Bitcoin (N) - (2010 to 2015)

91

Figure 9 Benchmark Portfolio (K) + Bitcoin (N) - (2010 to 2015) [To

Scale]

92

Figure 10 Benchmark Portfolio (K) + Bitcoin (N) – (2010 to 2015) [Short

Sale constraint relaxed]

93

Figures – Efficient Frontiers (2013 to 2014)

Figure 11 Benchmark Portfolio (K) + Gold (N) - (2013 to 2014)

94

Figure 12 Benchmark Portfolio (K) + Oil (N) - (2013 to 2014)

95

Figure 13 Benchmark Portfolio (K) + Euro (N) - (2013 to 2014)

96

Figure 14 Benchmark Portfolio (K) + JPY (N) - (2013 to 2014)

97

Figure 15 Benchmark Portfolio (K) + EMGBi (N) - (2013 to 2014)

98

Figure 16 Benchmark Portfolio (K) + Bitcoin (N) - (2013 to 2014)

99

Figures – Efficient Frontiers (2014 to 2015)

Figure 17 Benchmark Portfolio (K) + Gold (N) - (2014 to 2015)

100

Figure 18 Benchmark Portfolio (K) + Oil (N) - (2014 to 2015)

101

Figure 19 Benchmark Portfolio (K) + Euro (N) - (2014 to 2015)

102

Figure 20 Benchmark Portfolio (K) + JPY (N) - (2014 to 2015)

103

Figure 21 Benchmark Portfolio (K) + EMGBi (N) - (2014 to 2015)

104

Figure 22 Benchmark Portfolio (K) + Bitcoin (N) - (2014 to 2015) 105

67

Appendix 1 - Other Methods of Mean-variance spanning tests

The majority of academic studies, which evaluate portfolio allocation efficiency, tend to

utilise the methodologies proposed by Huberman and Kandel (1987) due to its simple

application and interpretation. Under the Huberman and Kandel (1987) model three

variations of the mean-variance spanning test are: the Wald Test (Hotelling, 1951)

(Anderson, 1984), the Likelihood Ratio (Gibbons et al., 1989) and the Lagrange Multiplier.

These three approaches test for spanning and tend to be calculated by utilising suitable

computer programs such as MatLab. Previous research from Berndt and Savin (1977) and

Breusch (1979) highlight that these test statistics have a definite relationship within finite

samples, as follows: Wald Test ≥ Likelihood Ratio ≥ Lagrange Multiplier.

Studies further investigating these asymptotic distribution-based statistics have demonstrated

that they have limited usefulness and meaning. Essentially, these statistics seem to have

some statistical significance, yet the wide spectrum of results tends to lead to conflicting

deductions concerning whether spanning is occurring or not, which in turn can result in the

false acceptance or rejection of the null hypotheses in finite samples; thus Kan and Zhou

(2012, pp.168) stress that “researchers need to be cautious in interpreting the p-values of

these tests”. Moreover, there are some general problems with the interpretation of what these

statistics actually represent, thus emphasising the need for tests which convey more

comprehensive results and provide improved quality of output and hence reliability of

interpretation. Essentially, the three aforementioned methods do not thoroughly explain why

the null hypothesis is either accepted or rejected in insolation, therefore limiting the actual

value of their application within empirical studies. Moreover, this F-test has been used to test

the spanning hypothesis in the literature for N = 1; it should be emphasised that this F-test is

only valid when N > 2 - this again leads to inconclusive figures. Therefore, although these

tests indicate some statistical significance, this does not necessarily imply that they hold any

form of economic significance.

More comprehensive and informative methods of conducting spanning tests have been

developed with inclusion of the stochastic discount factor. Academics such as Berkaert and

Urias (1996), Hansen and Jagannathan (1997) and Peñaranda and Sentana (2011) utilise this

new approach to explain the relationship between mean-variance frontiers and volatility

bounds. Although insightful, these further developments and utilisation of their

methodologies are beyond the scope of this particular study.

68

Appendix 2 – Bitcoin – Technical

The below diagram demonstrates in simple terms the fundamental applications Bitcoin is

designed for:

Figure 1: (Bitcoin Explained – How BTC Works, 2015)

Appendix 3 - Risks with Bitcoin

Due to Bitcoin’s under-regulation and pseudo-anonymous nature, some countries have

expressed their concern with the lack of regulation of Bitcoin and identified the

cryptocurrency as being too unstable for universal commercial use. For instance, the Chinese

government banned the usage of Bitcoin, even though Chinese interest in Bitcoin helped

accelerate Bitcoin to where it stands today (Gough, 2013).

Similarly, Stark (2013) reveals further that financial professionals tend to shun Bitcoin

simply due to its lack of liquidity, poor security record, and non-existent regulations and due

to the inability to mitigate exposures. On the same theme, Luther (2013) states that Bitcoin

will unlikely gain traction as a credible monetary system due to the underlying price

69

instability and lack of governmental acceptance. However, Cedillo (2013) confirms that

regulations tend to lag behind new technological innovations and take significant time to

fully incorporate and integrate into the system.

Most infamously, although designed to facilitate ethical and conventional business and

personal wealth transactions, the anonymous aspect of Bitcoin has attracted the likes of the

black market. The online store, ‘Silk-Road’ utilised and capitalised upon this characteristic

Bitcoin being an unregulated entity, to facilitate the drugs and weapons trade. However,

Greenberg (2013) details that only some 0.5% of the total Bitcoin traffic is associated with

the illicit operations of Silk Road. Not only this but numerous hacking of exchanges caused

the market to question Bitcoin’s credibility (see Appendix 4) as a widely adopted payment

method, thus causing panic and speculation leading to further volatility patterns.

Evidently, to speed up Bitcoin’s market-wide acceptance and progression, the Winklevoss

brothers have devised an innovative, regulated U.S bitcoin exchange called Gemini

(CoinDesk, 2014). Another expansion within the Bitcoin economy is the development of

over-the-counter (OTC) markets. Similar to the OTC markets available for exchange rates

and other assets proved in high demand; thus providing Bitcoin investors with security and

more options to profit. Moreover, the IRS (2014) now recognises Bitcoin as property for

taxation purposes. Yet this is yet to be fully incorporated into standard monetary practice.

Although this ground-breaking transaction technology is elegant in theory, its liability of

newness and general underrepresentation in academia, still causes a certain level of

underlying speculation. Thusly, this innovation could either be widely adopted by financial

practitioners and investors the world-over, or become just another modern day fad. Some

experts believe Bitcoin holds the potential to revolutionise the financial landscape for the

better, as loopholes in the technology are refined further. Furthermore, from an operational

perspective, Bitcoin is still relatively difficult to use for universal payments, leading to a

slowing down of adoption, thus characterising it as a niche innovation. Moreover,

economists are still largely sceptical of how viable the Bitcoin ideologies are, especially as

Bitcoin has come under the spotlight following high profile incidents. Generally, empirical

studies on Bitcoin within academia have been lacking; mainly due to Bitcoin’s relative

newness and volatile nature making it difficult to derive a considered understanding of the

unique dynamics that drive it. As such, Bitcoin’s random and volatile nature makes its price

trajectory challenging to measure and fully understand. Moreover, Segendorf (2014)

expresses that gauging and establishing the number of Bitcoin users is inherently difficult, if

not impossible due to Bitcoin’s international and pseudo-anonymous nature.

70

Appendix 4 - Mining and Blockchain Technology

The most important process that determines the Bitcoin supply, and therefore determining

the value of Bitcoin, is the mining process. In order to materialise Bitcoins, market players

need to solve highly complex algorithms through CPUs; this in turn helps facilitate and

verify transactions on the Blockchain, thus guaranteeing the mitigation of the risk of ‘double

spending’ (Brito et al., 2013). The ingenuity in the Bitcoin technology lies in the fact that

every transaction executed within the system is logged through a decentralised and

cryptographic network which records all transactions within a public ledger, also known as

the ‘Blockchain’.

This process gets increasingly more difficult to implement as the number of Bitcoins

materialised within the system increases. These algorithms become progressively more

complex and extensive to decipher as the Blockchain evolves and as ‘mining’ competition

intensifies. The mining process has seen huge levels of increasing global competition and

progression since Bitcoin’s inception; typically, mining is undertaken by huge networks of

computer systems at the forefront of processing power, thus making it extremely difficult to

join the mining race for the conventional individual investor with limited computational

resources. One significant issue lies in the fact that these mining organisations could

potentially pool computational resources, which effectively centralises the supply of Bitcoin

and therefore runs against the Bitcoin ideology. If enough pooling behaviour occurs, or if a

CPU is powerful enough to control 51% of the network, Bitcoins will effectively be

centrally-controlled – this flaw, although highly improbable, essentially goes against

Bitcoin’s decentralised ideology.

Interestingly, Meiklejohn (2013) finds that of the bitcoins mined in 2009-2010, more than

60% remain unspent or took more than one year to be spent. Moreover, research from Ron

and Shamir (2013) indicates that around 73% of Bitcoin wallet addresses only receive

Bitcoins and do not use them for further use. Likewise, Ratcliff (2014) documents that some

39% of Bitcoins are held for more than a year and 11% of all Bitcoins are left unused for

more than 4 years.

Side Chain Technologies are spin-off revisions of the Blockchain technology which have

been developed to undertake the underlying cryptographic framework to facilitate future

technologies. Currently developers such as ‘Ethereum’ are using Side Chain technologies to

develop a platform which allows the web applications to be fully decentralised. Available at:

https://www.ethereum.org/

71

Appendix 5 - Bitcoin Exchanges

As Bitcoin’s reputation as a credible payment method progressed, Bitcoin exchanges such as

Mt. Gox arose to facilitate the growth and streamline the transaction process of Bitcoins. The

success of Bitcoin was also accelerated by the fact that exchanges can occur 24/7 therefore

not restricting the owner to the limitations of exchange closing and opening times.

It was found however, that the concentration of these Bitcoin transactions made them a

hotspot for hacker attention and effort. Staggeringly, a study undertaken by Moore and

Christin (2013) demonstrated that some 45% of Bitcoin exchanges were ultimately

discontinued operations due to breaches of security, government intervention and excessive

traffic – only 20 of the original exchanges are in operation today. Their most profound

finding was that the levels of traffic through the exchange were highly correlated to the

chances of that exchange being a target for hackers and government intervention.

The most infamous of these exchanges was Mt. Gox; Abrams (2014) highlights that Mt Gox

was responsible for the loss of “744,000 of its customer’ bitcoins (worth approximately

US$300 million at the time of closure)”. In this same instance, the Winklevoss brothers, who

were famously affiliated with the foundation of Facebook, lost some $11million in April

2013 as Bitcoin’s price temporarily slumped (Popper and Lattman, 2013). As a result of this

negative press, Bitcoin developers devised continually robust technologies to securitise the

credibility of future exchanges.

Successful exchanges such as Kraken facilitate the buying, selling and trading of Bitcoin –

among other cryptocurrencies – to all global fiat currencies. The Kraken exchange can be

accessed from the following link: https://www.kraken.com/

Appendix 6 - Cyber, Regulatory, Political and Ethical Implications of Bitcoin Adoption

Jacobs (2011) explored the continuing legal and regulatory debate concerning the treatment

of Bitcoin in the U.S and European Union. Perhaps the most coverage on the applications of

Bitcoin is within the cryptographic and computer science literature. These papers are highly

specialist and are beyond the scope of this research however it is again vital that continual

research further progresses the underlying technologies to validate its wider acceptance.

Numerous event studies demonstrate how Bitcoin’s price reacts to certain announcements.

For instance Moore and Christan (2013) observe how Bitcoin exchange closures or breaches

72

of security effect Bitcoin’s price – finding that exchanges with lower trading volume are

more susceptible to suffer security breaches. Moreover, many industry reports, such as PwC

(2014) review the accounting, regulatory and tax implications of Bitcoin for institutional

investors. Although this area is still in its infancy and requires significantly more

investigation, this area is of high importance if investors are to consider Bitcoin as a store of

value or as a medium to transfer funds. Although these are highly important and debated

subject areas, they are beyond the scope of this study.

Appendix 7 - Other Cryptocurrencies

To date there are around 530 actively used cryptocurrencies in operation, although only 10 of

these currencies have a market capitalisation over $10 million – the most notable of which

are highlighted below (As of 28 February 2015):

Figure 2: (Cryptocurrency Market Capitalisation, 2015)

Interestingly, Bitcoin has inspired new breeds of cryptocurrency which aim to not only

capture market value, but to refine some of the fundamental flaws with the Bitcoin

technology. For instance, some academics believe that the maximum fixed supply of 21

million may hinder Bitcoin’s long-term success; to address this issue Litecoin aims to enable

a fixed supply of 84 million units. Alternatively, Dogecoin has an infinite supply, which in

effect, helps to remedy the deflationary issues associated with Bitcoin (Krugman, 2011).

Rank Name Market Cap Price Available Supply

1 Bitcoin $ 3,512,149,361 $ 252.87 13,889,150 BTC

2 Ripple* $ 417,631,887 $ 0.013088 31,908,551,587 XRP

3 Litecoin $ 67,686,840 $ 1.84 36,862,254 LTC

4 BitShares* $ 26,932,944 $ 0.010766 2,501,643,489 BTS

5 Darkcoin $ 16,437,061 $ 3.18 5,162,572 DRK

6 Dogecoin $ 14,310,754 $ 0.000145 98,429,436,308 DOGE

* Not Mineable

73

Appendix 8 - Behavioural Finance and Bitcoin

The extremeness of Bitcoin’s price fluctuation, and the individuals associated with trading it,

has inspired a handful of scholars to empirically investigate this area from a behavioural

finance perspective.

Firstly, Ciaian et al. (2014, pp.6) documents that there are “several important factors which

affect the behaviour of Bitcoin investors in addition to the traditional ones”. Interestingly,

Wu (2014) conveys that those affiliated with Bitcoin almost “feel part of a growing digital

movement” or new order moving against government and corporate control.

Correspondingly, Bradbury (2013) raises the issue that individuals that possess and trade

with Bitcoin generally “express an opposition to the traditional financial sector that lost their

trust in the recent financial crisis”. Another notable stigma attached to Bitcoins is conveyed

by Raskin (2013) who details that “early Bitcoin users often described themselves as

libertarians, distrusting governments generally and monetary policy specifically”.

Perhaps the most robust explanation of Bitcoin activity lends from the respected works of

Grullon et al. (2004) and Barber and Odean (2008), who provide strong empirical evidence

to suggest that new investor preferences are highly skewed by their tendency to have limited

attention to specific information leading to highly irrational and sub-optimal behaviour.

Further studies convey that new investors are liable to be distorted by limited attention for

alternative investments. New investors generally have a preference for investment

opportunities lauded by the media. For instance, Lee (2014) finds investors are highly

sensitive to positive and negative news stories. Specifically, Lee (2014) utilises Google SVI

methodology and tests, which has been proven to direct represent investor attention: thus the

study find a highly significant correlation between investor attention and Bitcoin price

dynamics. From this basis, further academic research was able to conduct acute tests to

gauge whether Bitcoin prices are overinflated and driven by investor sentiment during

specific periods. Similar findings were also demonstrated by Kristoufek (2013), who

examined the connection of Google Trends data and Wikipedia activity to the price of

Bitcoin (see Chapter 2.5.).

Moreover, Husler et al. (2013) examines the emergence of bubbles that exhibit faster-than-

exponential growth. The bubble and crash of Bitcoin in April of 2013 is mentioned as such

an example. The study utilises a learning-to-forecast laboratory experiment with human

subjects and concludes that these types of super-exponential bubbles can occur in such a

setting. In fact, a common feature of such bubbles is found to be that prices are only loosely

74

connected to fundamentals. This study helps to understand how the dramatic price swings

have been possible because Bitcoin is completely disconnected from fundamentals.

Similarly, Blundell-Wignall (2014) conveys that the behaviour of investors in Bitcoin

generally fits the ‘greater fool theory’; extending investor valuations bias towards a

continually growing trajectory which partly provides a behavioural explanation for the rapid

emergence of over-inflated prices.

Evidently, these instances add to the complexity of Bitcoin and generally make us question

whether Bitcoin will ever become a widely accepted as a viable investment medium, or

whether it exclusively acts as a symbol against bureaucracy, attracting the likes of irrational

investors. Although this area of finance would be of great interest and significance for the

furtherance of behavioural finance research, empirically testing this would be inherently

difficult to conduct in practice. Moreover, although individual Bitcoin investors and their

inherently irrational behaviour has been lightly touched upon, studies evaluating institutional

investor treatment of Bitcoin is largely under-researched.

75

8. Glossary

Glossary 1 - Portfolio Theory - formulae

Return

For a set of risky assets and associated weights, the formula of expected return for n assets is

denoted as:

1

( )n

P i i

i

E r w E r

when:

1

n

i

i

w

= 1.0

n = the no. of securities;

iw

= the percentage invested in risk asset i;

,i Pr r

= The total return on ith security and the total portfolio p

E

= The expected result of the calculated terms

Risk

Fundamentally, the variance of the individual assets is the sum of squared deviations from

the mean. Thus, the squared root of the variance gives the standard deviation.

Therefore, the portfolio variance is equal to the weighted-average covariance of the returns

on the individual assets:

2

1 1

Var Cov ,n n

p p i j i j

i j

r w w r r

Alternatively, the covariance can be denoted with respect to the correlation co-efficient as

follows:

Cov ,i j ij i j ijr r

Where ijis the correlation between the return on asset i, and the return on asset j. The

standard deviation of i and j are represented by σi and σj respectively.

1 1

Varn n

p i j ij i j

i j

r w w

76

Glossary 2 - Risk free rate

The risk-free rate (denoted as rf) is a theoretical rate of return which assumes that an

investment has no risk of financial loss attached. Risk-averse investors are assumed to have a

preference to invest at the risk-free rate as it has little to no underlying risk. Essentially,

triple AAA rated government bonds such as the UK and the US are deemed as a proxy for

the risk-free rate, although in real terms, some underlying default risk is present. Fama and

French (2004, pp25) define the risk-free rate as the rate of which “clears the market for

borrowing and lending”. As such, the risk-free rate assumption is commonly incorporated

within highly cited models such as the CAPM.

As of February 2015 the yield of a 30 Year UK Gov. Bond was 2.5% (ft.com, 2015), for the

empirical analysis this rate is transformed into a weekly risk free rate of return of

0.048077%. This risk-free rate figure is a key component in determining the Sharpe ratios/

tangency portfolio.

Glossary 3 – Sharpe Ratios

The Sharpe Ratio, devised by William F. Sharpe (1966), is the most widely used measure to

compare the performance of a set of portfolios. Essentially, the Sharpe Ratio is the average

return in excess of the risk free rate (rf) per unit of portfolio risk; i.e. the higher the Sharpe

Ratio, the greater the portfolio performance.

Figure 10: (Smith, 2012)

By subtracting the rf rate from the average return, the risk-taking activities undertaken can be

observed in isolation. Hence, a Sharpe ratio above ‘0’ achieves a greater return than the risk-

free rate (see Glossary 2). Note, however, that the Sharpe Ratio has limited accuracy if

portfolio expected returns do not follow the normal distribution.

77

9. Tables

Table 11 - Kan and Zhou (2012) Step-down Tests

The above comparison matrix presents the results of all of the Kan and Zhou (2012) step-

down approach mean-variance spanning tests using the benchmark portfolio (K) and the

each of the respective test assets. The first box presents the results over the period of 2010 to

2015; the second box 2014 to 2015; third box from 2013 to 2014 and the final box presents

the results of the spanning test when all test assets (N) were collectively added into a

portfolio and tested against the benchmark portfolio (K). The step-down test, where F1 is an

F-test of α = 0N and the F2 is an F-test of δ = 0N conditional of α = 0N. One asterisk joint to

the p-values highlights significance to 5% and two asterisks denotes significance at a 10%

level. Note all p-values are exact under the normality assumption of the residuals. If the F1

and associative p-value are significant to 5%, the tangency portfolio can be improved. If the

F2 test is rejected at a 5%, the global mean-variance can be improved.

Comparison Matrix

All - (2010 to 2015)

F1 p-value F2 p-value

Gold 0.711 0.3999 14.398 0.000189*

Oil 0.926 0.3368 0.083 0.7731

Euro 2.388 0.1237 31.741 5.124E-08*

JPY 5.347 0.0216* 18.059 0.000031*

EMG BI 0.304 0.5822 0.620 0.4317

BTC 12.105 0.0006* 2.085 0.1501

2014 to 2015

F1 p-value F2 p-value

Gold 1.539 0.2210 3.2610 0.0774**

Oil 1.030 0.3155 1.5817 0.2147

Euro 2.427 0.1261 0.6615 0.4201

JPY 8.367 0.00582** 4.3827 0.0417*

EMG BI 1.873 0.1778 2.2979 0.1362

BTC 0.300 0.5862 0.0357 0.8508

2013 to 2014

F1 p-value F2 p-value

Gold 3.232 0.0788** 2.898 0.0953

Oil 1.061 0.3084 0.071 0.7910

Euro 0.499 0.4834 15.837 0.000238*

JPY 0.726 0.3988 0.879 0.3533

EMG BI 1.582 0.2148 3.268 0.077**

BTC 6.973 0.0113* 3.646 0.0623**

2010 to 2015

F1 p-value F2 p-value

All Test Assets (N) 20.03 1.20209E-05 55.6211419 1.7835E-12

78

Table 12 – Constituents of Optimal Portfolios

The above table presents the weights of each of the optimal portfolios constructed when the

benchmark assets (K) were combined with each of the test assets (N) for each of the time

periods evaluated. Also shown above are the respective Sharpe Ratios, mean and standard

deviation of each of the optimal portfolios. If the constituent weightings do not differ from

the benchmark portfolio (K), the tests asset (N) was of no added benefit to the portfolio.

Hence, if the portfolio (K + N) differs from the benchmark (K), the test asset (N) enhances

the expected return or assists with reducing portfolio risk. Note that these optimal

constituents are represented by the green triangle in the following efficient frontiers.

2010 to 2015 FTSE All-Share UK GILTS UK 10-Y G BI USD to UK FTSE WLD Test Asset (N) Sharpe Ratio Mean Std. Dev.

Benchmark (K) 14.96% 78.58% 6.46% 0.1528 0.145% 0.631%

Gold 9.66% 77.69% 12.65% 0.1528 0.145% 0.631%

Oil 14.95% 78.58% 6.47% 0.1528 0.145% 0.631%

Euro 13.40% 72.51% 6.80% 7.29% 0.1540 0.139% 0.587%

JPY 9.52% 66.41% 2.60% 21.46% 0.1878 0.139% 0.483%

EMGBi 14.96% 78.58% 6.46% 0.1528 0.145% 0.631%

Bitcoin 16.78% 77.16% 1.18% 4.89% 0.2868 0.402% 1.234%

2013 to 2014 FTSE All-Share UK GILTS UK 10-Y G BI USD to UK FTSE WLD Test Asset (N) Sharpe Ratio Mean Std. Dev.

Benchmark (K) 51.40% 48.60% 0.1178 0.156% 0.915%

Gold 51.40% 48.60% 0.1178 0.156% 0.915%

Oil 51.40% 48.60% 0.1178 0.156% 0.915%

Euro 48.06% 40.74% 11.20% 0.1184 0.150% 0.860%

JPY 30.64% 3.58% 65.79% 0.2332 0.313% 1.135%

EMGBi 51.40% 48.60% 0.1178 0.156% 0.915%

Bitcoin 49.99% 42.42% 7.59% 0.4428 0.950% 2.037%

2014 to 2015 FTSE All-Share UK GILTS UK 10-Y G BI USD to UK FTSE WLD Test Asset (N) Sharpe Ratio Mean Std. Dev.

Benchmark 1.85% 73.57% 24.57% 0.4933 0.333% 0.577%

Gold 1.85% 73.57% 24.57% 0.4933 0.333% 0.577%

Oil 1.85% 73.57% 24.57% 0.4933 0.333% 0.577%

Euro 60.69% 18.22% 21.09% 0.5230 0.303% 0.488%

JPY 66.52% 14.83% 18.65% 0.5648 0.282% 0.414%

EMGBi 1.85% 73.57% 24.57% 0.4933 0.333% 0.577%

Bitcoin 1.85% 73.57% 24.57% 0.4933 0.333% 0.577%

FTSE All-Share UK GILTS Euro JPY BTC Sharpe Ratio Mean Std. Dev.

All Test Assets 10.59% 65.22% 1.71% 19.42% 3.06% 0.3017 0.299% 0.832%

79

Table 13 - Correlation Co-efficient Matrix – 2010 to 2015

The matrix below represents the correlations between the benchmark assets (K) and the test assets (N) respectively over the entire test period from 2010

to 2015.

FTSE All-Share UK GILTS UK 10-Yr Gov BI USD to UK FTSE Wld REIT Gold Brent Oil EURO to UK JPY to UK EMG BI BTC

FTSE All-Share 1.0000 -0.3897 -0.3446 0.1904 0.7570 -0.0291 0.4245 -0.0984 0.2400 0.4218 0.0622

UK GILTS -0.3897 1.0000 0.9425 -0.2023 -0.1438 0.3190 -0.2510 0.0088 -0.4450 0.1929 0.0258

UK 10-Yr Gov BI -0.3446 0.9425 1.0000 -0.1847 -0.1067 0.3068 -0.2192 0.0600 -0.4759 0.2783 0.0265

USD to UK 0.1904 -0.2023 -0.1847 1.0000 0.3137 0.2555 0.3469 0.1499 0.5510 0.2357 0.0754

FTSE Wld REIT 0.7570 -0.1438 -0.1067 0.3137 1.0000 0.0829 0.3489 -0.1952 0.1635 0.4656 0.1083

Gold -0.0291 0.3190 0.3068 0.2555 0.0829 1.0000 0.1495 -0.0671 -0.0341 0.2551 0.1054

Brent Oil 0.4245 -0.2510 -0.2192 0.3469 0.3489 0.1495 1.0000 -0.0282 0.2079 0.1924 0.0608

EURO to UK -0.0984 0.0088 0.0600 0.1499 -0.1952 -0.0671 -0.0282 1.0000 0.1415 -0.0229 -0.0435

JPY to UK 0.2400 -0.4450 -0.4759 0.5510 0.1635 -0.0341 0.2079 0.1415 1.0000 0.1071 0.0599

EMG BI 0.4218 0.1929 0.2783 0.2357 0.4656 0.2551 0.1924 -0.0229 0.1071 1.0000 0.0389

BTC 0.0622 0.0258 0.0265 0.0754 0.1083 0.1054 0.0608 -0.0435 0.0599 0.0389 1.0000

80

Table 14 - Correlation Co-efficient Matrix – 2013 to 2014

The matrix below represents the correlations between the weekly returns of the benchmark assets (K) and the test assets (N) respectively over the sub-

period test from 2013 to 2014.

FTSE All-Share UK GILTS UK 10-Yr Gov BI USD to UK FTSE Wld REIT Gold Brent Oil EURO to UK JPY to UK EMG BI BTC

FTSE All-Share 1.0000 0.1535 0.2426 -0.1607 0.6299 -0.0102 0.3418 -0.0662 -0.0155 0.5354 -0.1163

UK GILTS 0.1535 1.0000 0.9611 0.1556 0.5462 0.3946 -0.0500 -0.0300 -0.1615 0.6159 0.2296

UK 10-Yr Gov BI 0.2426 0.9611 1.0000 0.1684 0.5767 0.3451 0.0210 0.0586 -0.2499 0.6302 0.1407

USD to UK -0.1607 0.1556 0.1684 1.0000 0.1178 0.3086 0.2274 0.5362 0.2588 0.2138 0.0400

FTSE Wld REIT 0.6299 0.5462 0.5767 0.1178 1.0000 0.1510 0.0949 -0.0840 0.0849 0.7538 0.1065

Gold -0.0102 0.3946 0.3451 0.3086 0.1510 1.0000 0.2891 0.0761 -0.0243 0.2096 0.2536

Brent Oil 0.3418 -0.0500 0.0210 0.2274 0.0949 0.2891 1.0000 0.1627 0.0016 0.0965 -0.0655

EURO to UK -0.0662 -0.0300 0.0586 0.5362 -0.0840 0.0761 0.1627 1.0000 0.1331 -0.0888 0.0904

JPY to UK -0.0155 -0.1615 -0.2499 0.2588 0.0849 -0.0243 0.0016 0.1331 1.0000 0.0822 0.3735

EMG BI 0.5354 0.6159 0.6302 0.2138 0.7538 0.2096 0.0965 -0.0888 0.0822 1.0000 0.0358

BTC -0.1163 0.2296 0.1407 0.0400 0.1065 0.2536 -0.0655 0.0904 0.3735 0.0358 1.0000

81

Table 15 - Descriptive Statistics – 2013 to 2014

The following table highlights the descriptive statistics of the weekly returns of the benchmark assets (K) and the test assets (N) over the sub-period of

2013 to 2014:

FTSE All-Share UK GILTS UK 10-Yr Gov BIUSD to UK FTSE Wld REITGold Brent Oil EURO to UK JPY to UK EMG BI BTC

Mean 0.0013199 -0.0000585 -0.0006766 0.0010821 -0.0012484 -0.0048043 -0.0007866 0.0007623 0.0035080 -0.0015350 0.1025321

Standard Error 0.0023169 0.0011022 0.0013833 0.0013747 0.0029242 0.0041826 0.0031908 0.0010908 0.0021387 0.0023387 0.0351358

Median 0.0028692 -0.0005250 0.0002904 0.0020162 0.0028741 -0.0007433 -0.0001779 0.0014868 0.0044684 -0.0015757 0.0710259

Standard Deviation 0.0167076 0.0079480 0.0099753 0.0099134 0.0210865 0.0301615 0.0230092 0.0078659 0.0154224 0.0168643 0.2533681

Sample Variance 0.0002791 0.0000632 0.0000995 0.0000983 0.0004446 0.0009097 0.0005294 0.0000619 0.0002378 0.0002844 0.0641954

Kurtosis 1.3911137 1.3904495 3.5371906 -0.3502394 1.1683069 5.8535605 0.3162464 -0.8438413 3.6724917 7.0571618 3.7413616

Skewness -0.2458188 -0.5644472 -1.1077857 -0.4199114 -0.9918884 -1.6527859 -0.6000463 -0.0707766 0.8274881 -1.0849990 1.1650955

Range 0.0940537 0.0423512 0.0578722 0.0414470 0.1013407 0.1785913 0.1045019 0.0325340 0.0959545 0.1214359 1.5880045

Minimum -0.0468404 -0.0268864 -0.0391937 -0.0239026 -0.0664593 -0.1344211 -0.0607919 -0.0141482 -0.0322278 -0.0751286 -0.5605867

Maximum 0.0472132 0.0154648 0.0186785 0.0175444 0.0348813 0.0441702 0.0437100 0.0183859 0.0637267 0.0463072 1.0274179

Sum 0.0686333 -0.0030410 -0.0351825 0.0562684 -0.0649150 -0.2498258 -0.0409057 0.0396377 0.1824163 -0.0798201 5.3316712

Count 52 52 52 52 52 52 52 52 52 52 52

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Table 16 - Correlation Co-efficient Matrix – 2014 to 2015

The matrix below represents the correlations between the weekly returns of the benchmark assets (K) and the test assets (N) respectively over the sub-

period test from 2014 to 2015.

FTSE All-Share UK GILTS UK 10-Yr Gov BI USD to UK FTSE Wld REIT Gold Brent Oil EURO to UK JPY to UK EMG BI BTC

FTSE All-Share 1.0000 -0.4612 -0.4136 0.0573 0.6120 -0.2059 0.2672 0.4336 0.5058 0.7119 -0.0887

UK GILTS -0.4612 1.0000 0.9502 -0.0536 -0.0339 0.4579 -0.4011 -0.1559 -0.5673 -0.2915 0.1093

UK 10-Yr Gov BI -0.4136 0.9502 1.0000 -0.0973 0.0100 0.3821 -0.3415 -0.2500 -0.5862 -0.1823 0.1680

USD to UK 0.0573 -0.0536 -0.0973 1.0000 0.1474 0.3258 0.3702 0.4303 0.4019 0.2335 -0.0706

FTSE Wld REIT 0.6120 -0.0339 0.0100 0.1474 1.0000 0.0950 0.0609 0.2181 0.1039 0.6793 -0.0858

Gold -0.2059 0.4579 0.3821 0.3258 0.0950 1.0000 -0.1092 -0.0395 -0.3684 -0.1218 -0.0481

Brent Oil 0.2672 -0.4011 -0.3415 0.3702 0.0609 -0.1092 1.0000 0.2721 0.3224 0.4015 0.0254

EURO to UK 0.4336 -0.1559 -0.2500 0.4303 0.2181 -0.0395 0.2721 1.0000 0.4180 0.3013 -0.0183

JPY to UK 0.5058 -0.5673 -0.5862 0.4019 0.1039 -0.3684 0.3224 0.4180 1.0000 0.3412 -0.1509

EMG BI 0.7119 -0.2915 -0.1823 0.2335 0.6793 -0.1218 0.4015 0.3013 0.3412 1.0000 -0.0770

BTC -0.0887 0.1093 0.1680 -0.0706 -0.0858 -0.0481 0.0254 -0.0183 -0.1509 -0.0770 1.0000

83

Table 17 - Descriptive Statistics – 2014 to 2015

The following table highlights the descriptive statistics of the weekly returns of the benchmark assets (K) and the test assets (N) over the sub-period of

2013 to 2014

FTSE All-Share UK GILTS UK 10-Yr Gov BI USD to UK FTSE Wld REIT Gold Brent Oil EURO to UKJPY to UK EMG BI BTC

Mean 0.001057 0.002860 0.002347 -0.001821 0.004894 0.000385 -0.015328 0.001922 0.001024 0.001462 -0.015235

Standard Error 0.002606 0.000879 0.000942 0.001091 0.001993 0.002648 0.004535 0.001116 0.001838 0.001437 0.015049

Median 0.003576 0.002417 0.002262 -0.000098 0.004309 0.001873 -0.010143 0.001223 0.000344 0.002428 -0.023061

Standard Deviation 0.018796 0.006341 0.006795 0.007868 0.014374 0.019094 0.032705 0.008047 0.013252 0.010363 0.108522

Sample Variance 0.000353 0.000040 0.000046 0.000062 0.000207 0.000365 0.001070 0.000065 0.000176 0.000107 0.011777

Kurtosis 3.953912 -0.488349 -0.510080 1.790085 2.638693 0.110942 1.696114 0.742364 2.736740 6.783695 1.467041

Skewness -0.408360 0.192789 0.174769 -0.509421 -0.511651 -0.449144 -1.272209 0.412632 0.750051 -0.700627 0.818558

Range 0.128177 0.026306 0.030019 0.046413 0.086989 0.084227 0.145457 0.041065 0.076577 0.078019 0.527451

Minimum -0.067773 -0.010457 -0.011893 -0.027265 -0.046668 -0.048397 -0.110609 -0.014976 -0.026650 -0.041403 -0.217937

Maximum 0.060404 0.015849 0.018126 0.019148 0.040321 0.035830 0.034848 0.026090 0.049927 0.036616 0.309514

Sum 0.054979 0.148742 0.122066 -0.094686 0.254463 0.020015 -0.797041 0.099961 0.053245 0.076048 -0.792223

Count 52 52 52 52 52 52 52 52 52 52 52

84

10. Figures

Figure 1 – Benchmark Assets (K) + All Test Assets (N) - (2010 to 2015)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2010 to 2015. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test assets (N): Gold,

Oil, Euro, Japanese Yen, Emerging Market Bond Index and Bitcoin. The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-

variability ratio). The blue Capital Allocation Line (CAL) represents all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK

government bond risk free rate equalling 2.5% per annum. Finally the mean and standard deviation of the benchmark assets (K) are represented by the red diamonds and the

coloured points are the risky test assets (N). Note that the scale of both the x and y-axes has been adjusted to allow the incorporation of Bitcoin.

85

Figure 2 – Benchmark Assets (K) + All Test Assets (N) – (2010 to 2015) [Short Sales constraint relaxed]

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2010 to 2015. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test assets (N): Gold,

Oil, Euro, Japanese Yen, Emerging Market Bond Index and Bitcoin. The Green line represents the efficient frontier of the benchmark asset portfolio (K) plus the test assets

(N): Gold, Oil, Euro, Japanese Yen, Emerging Market Bond Index and Bitcoin when short sales are permitted. Note that the scale of both the x and y-axes has been adjusted

to allow the incorporation of Bitcoin.

86

Figure 3- Benchmark Portfolio (K) + Gold (N) – (2010 to 2015)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2010 to 2015. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N): Gold.

The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital Allocation Line (CAL) represents

all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling 2.5% per annum. Finally the mean and

standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes gold (N).

87

Figure 4- Benchmark Portfolio (K) + Oil (N) – (2010 to 2015)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2010 to 2015. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N): Oil. The

green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital Allocation Line (CAL) represents all of

the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling 2.5% per annum. Finally the mean and

standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes Brent Oil (N).

88

Figure 5- Benchmark Portfolio (K) + Euro (N) – (2010 to 2015)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2010 to 2015. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N): Euro.

The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital Allocation Line (CAL) represents

all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling 2.5% per annum. Finally the mean and

standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes the Euro (N).

89

Figure 6 – Benchmark Portfolio (K) + JPY (N) - (2010 to 2015)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2010 to 2015. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N): Japanese

Yen. The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital Allocation Line (CAL)

represents all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling 2.5% per annum. Finally

the mean and standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes the Japanese Yen.

90

Figure 7 – Benchmark Portfolio (K) + EMGBi (N) - (2010 to 2015)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2010 to 2015. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N):

Emerging Market Bond Index. The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital

Allocation Line (CAL) represents all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling

2.5% per annum. Finally the mean and standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes the Emerging

Market Bond Index.

91

Figure 8 – Benchmark Portfolio (K) + Bitcoin (N) - (2010 to 2015)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2010 to 2015. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N): Bitcoin.

The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital Allocation Line (CAL) represents

all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling 2.5% per annum. Finally the mean and

standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes Bitcoin (N). Note that the scale of both the x and y-axes

has been adjusted to allow the incorporation of Bitcoin.

92

Figure 9 – Benchmark Portfolio (K) + Bitcoin (N) - (2010 to 2015) – [To Scale]

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2010 to 2015. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N): Bitcoin.

The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital Allocation Line (CAL) represents

all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling 2.5% per annum. Finally the mean and

standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes Bitcoin (N).

93

Figure 10 - Benchmark Portfolio (K) + Bitcoin (N) – (2010 to 2015) [Short Sales constraint relaxed]

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2010 to 2015. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index when short and long sales are permitted. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously

described] plus the test asset (N): Bitcoin when short and long sales are permitted. The green line denotes the efficient frontier of the benchmark portfolio (K) plus the

incorporation of Bitcoin (N) when only long sales can only take place (see Figure 8).

94

Efficient Frontiers (2013 to 2014)

Figure 11 – Benchmark Portfolio (K) + Gold (N) - (2013 to 2014)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2013 to 2014. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N): Gold.

The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital Allocation Line (CAL) represents

all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling 2.5% per annum. Finally the mean and

standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes gold (N).

95

Figure 12 – Benchmark Portfolio (K) + Oil (N) - (2013 to 2014)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2013 to 2014. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N): Oil. The

green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital Allocation Line (CAL) represents all of

the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling 2.5% per annum. Finally the mean and

standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes Brent Oil (N).

96

Figure 13 – Benchmark Portfolio (K) + Euro (N) - (2013 to 2014)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2013 to 2014. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N): The

Euro. The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital Allocation Line (CAL)

represents all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling 2.5% per annum. Finally

the mean and standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes the Euro (N).

97

Figure 14 – Benchmark Portfolio (K) + JPY (N) - (2013 to 2014)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2013 to 2014. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test assets (N): The

Japanese Yen. The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital Allocation Line

(CAL) represents all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling 2.5% per annum.

Finally the mean and standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes the Japanese Yen (N).

98

Figure 15 – Benchmark Portfolio (K) + EMGBi (N) - (2013 to 2014)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2013 to 2014. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N):

Emerging Market Bond Index. The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital

Allocation Line (CAL) represents all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling

2.5% per annum. Finally the mean and standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes the Emerging

Market Bond Index (N).

99

Figure 16 – Benchmark Portfolio (K) + Bitcoin (N) - (2013 to 2014)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2013 to 2014. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N): Bitcoin.

The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital Allocation Line (CAL) represents

all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling 2.5% per annum. Finally the mean and

standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes Bitcoin (N). Note that the scale of both the x and y-axes

has been adjusted to allow the incorporation of Bitcoin.

100

Efficient Frontiers (2015 to 2015)

Figure 17 - Benchmark Portfolio (K) + Gold (N) - (2014 to 2015)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2014 to 2015. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N): Gold.

The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital Allocation Line (CAL) represents

all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling 2.5% per annum. Finally the mean and

standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes gold (N).

101

Figure 18 – Benchmark Portfolio (K) + Oil (N) - (2014 to 2015)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2014 to 2015. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N): Oil. The

green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital Allocation Line (CAL) represents all of

the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling 2.5% per annum. Finally the mean and

standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes Brent Oil (N).

102

Figure 19 – Benchmark Portfolio (K) + Euro (N) - (2014 to 2015)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2014 to 2015. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N): Euro.

The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital Allocation Line (CAL) represents

all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling 2.5% per annum. Finally the mean and

standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes the Euro (N).

103

Figure 20 – Benchmark Portfolio (K) + JPY (N) - (2014 to 2015)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2014 to 2015. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N): Japanese

Yen. The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital Allocation Line (CAL)

represents all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling 2.5% per annum. Finally

the mean and standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes the Japanese Yen (N).

104

Figure 21 – Benchmark Portfolio (K) + EMGBi (N) - (2014 to 2015)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2014 to 2015. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N):

Emerging Market Bond Index. The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital

Allocation Line (CAL) represents all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling

2.5% per annum. Finally the mean and standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes the Emerging

Market Bond Index (N).

105

Figure 22 - Benchmark Portfolio (K) + Bitcoin (N) - (2014 to 2015)

The above figure demonstrates the mean-variance frontier devised by the methodology described in chapter 3 from 2014 to 2015. The red line represents the mean-variance

frontier of the benchmark portfolio (K) for the UK-centric conservative portfolio, which includes the FTSE All-Share, UK Gilts, 10-Year UK Government Bonds, US Dollar

and FTSE Real Estate World Index. The blue line represents the efficient frontier of the benchmark asset portfolio (K) [previously described] plus the test asset (N): Bitcoin.

The green triangle represents the optimal risky portfolio/portfolio with the highest tangency (reward-to-variability ratio). The blue Capital Allocation Line (CAL) represents

all of the possible combinations of risk-free and risky assets – the y-intercept is the 30-Yr UK government bond risk free rate equalling 2.5% per annum. Finally the mean and

standard deviation of the benchmark assets (K) are represented by the red diamonds and the yellow square denotes Bitcoin (N).