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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
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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.
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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
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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.
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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
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8.
Glossary 75
9.
Tables 77
10. Figures 84
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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.
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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
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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.
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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).