8/15/2018
MIF Thesis for 2017-18
An Examination of
Cryptocurrencies as a Means
for Portfolio Diversification
Universiteit van Amsterdam
Student Name: Yuchen Jin
Student ID: 11935324
Supervisor: Torsten Jochem
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Statement of Originality
This document is written by student Yuchen Jin who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in
creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
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Abstract
The emergence of cryptocurrency has provided an alternative investment tool to investors. The value of cryptocurrency does not depend on the financial market but closely links to the blockchain. The low correlation with financial markets makes it an ideal tool for portfolio diversification. Contrary, the exceptional return of cryptocurrencies has great uncertainty. This may cause disruptive effects on an existing portfolio. We aim to examine how and to what extent cryptocurrency will work in portfolio diversification. We form a well-diversified crypto portfolio with optimal allocation and another portfolio consisting of nine traditional financial instruments with optimal allocation based on the Markowitz’s theory of mean-variance optimization. By adding the crypto fund to the traditional portfolio, we generate a new optimal portfolio of crypto assets as well as traditional assets. We find that the new portfolio exhibits significantly better risk-reward trade-off than the traditional portfolio. Furthermore, we conduct a bootstrapping simulation to enhance the robustness of our experiment. We hold the opinion that cryptocurrency can play an active role in portfolio diversification. The optimal allocation of cryptocurrency in a diversified portfolio is approximately 30%.
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Contents
Statement of Originality ... 1
Abstract ... 2
Ⅰ. Introduction ... 4
II. Literature Overview and Preliminary Research ... 6
2.1 Studies of Cryptocurrencies ... 6
2.2 Cryptocurrencies as a Tool for Diversification ... 7
2.3 Crypto Funds ... 8
Ⅲ. Data and Methodology ... 10
3.1 Data Collection ... 10
3.2 Methodology ... 13
Ⅳ. Empirical Results... 20
4.1 Descriptive Statistics ... 20
4.2 Correlations between cryptocurrencies and other financial assets ... 24
4.3 Evaluation of the optimal portfolio ... 25
4.4 A prediction of the expected return and risk ... 27
4.5 The solution to the central questions ... 28
Ⅴ. Conclusion... 30
References ... 32
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I. Introduction
The emergence of cryptocurrency has provided an alternative money market instrument for investors. This digital asset depends on Blockchain, which is a digital bookkeeping system (Yermack 2013). Unlike conventional currencies, cryptocurrencies are fully decentralized without regulations of central authority (Dwyer 2015). The lack of centralization removes the potential of intervention from a regulatory party and lowers the transaction fee. Therefore, cryptocurrency has been considered as a smart alternative investment and has become appealing to a wide variety of investors. Currently, there are more than 4,000 cryptocurrencies in existence and the number is still expected to grow (CoinLib 2018). Among these existing ones, Bitcoin, Ethereum, and Ripple are most popular in terms of market capitalization.
The cryptocurrency market has witnessed the price soaring and market capitalization incredibly accumulating in recent years. Taking Bitcoin as an example, from May 2015 to May 2018, the price of the oldest and most popular cryptocurrency, jumped from USD 232 to USD 9,655, while the market capitalization of Bitcoin increased from 3.3 to 167.5 billion (CoinMarketCap 2018). In the meantime, volatility of cryptocurrencies remains at a high level despite high rates of return. According to CoinMarketCap (2018), the value of Bitcoin dropped significantly from USD 10,951 on March 01 2018 to USD 6,844 on April 01 2018 but then bounced to 9,700 at the beginning of May 2018. Evidence shows that the return of cryptocurrency is quite volatile and crypto investors are speculative (Corbet et.al 2018). Consequently, it casts doubt on whether it is a reliable alternative investment vehicle. On one hand, cryptocurrency is relatively isolated from traditional financial assets (Meegan et.al 2018). Bouri et al.’s research (2017) provides evidence that there were insignificant correlations between Bitcoin and traditional asset classes, such as stock indices, commodity indices, bonds and precious metals. As a result, Bitcoin can act as a diversification tool to some extent. On the other hand, cryptocurrencies might be disruptive to an existing portfolio because it will increase the portfolio volatility. The risks associated with cryptocurrencies reduces the attraction of them for diversification purposes. Therefore, it is essential to examine the overall effect of the two features to detect the usefulness of diversification.
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This paper aims to examine whether cryptocurrency can serve as an effective means of portfolio diversification or not. Instead of investing in individual cryptocurrency (e.g. Bitcoin), it will focus on several popular cryptocurrencies. Basically, we design experiments to compare the performance of a crypto portfolio, a portfolio of several other financial assets and a portfolio including both crypto assets and other financial assets. It starts with the construction of a crypto fund consisting of four mainstream cryptocurrencies, namely Bitcoin, Ethereum, Ripple, and Litecoin, which have a larger market capitalization than the remaining cryptocurrencies in circulation. It will calculate the daily rate of return and volatility of these cryptocurrencies from 2015 to 2018. Then, it will build a variance-covariance matrix of the cryptocurrencies and calculate the optimal weights of each component according to Markowitz’s theory of Mean-Variance Optimization. For comparison, we will create an optimal portfolio consisting of several traditional financial assets with the same method. Those asset classes include major stock indices (e.g. MSCI and S&P 500), high-yield bond, general commodity index, precious metals, as well as private equities. We hypothesize that crypto assets will significantly affect a portfolio with traditional financial assets. Thus, a new variance-covariance matrix will be constructed not only to examine the correlation between the crypto fund and the other assets, but also to create a new optimized portfolio incorporating both the crypto fund and the other traditional financial instruments. In terms of risk estimation, we will apply the bootstrapping method in predicting the portfolio return and volatility. By conducting these experiments, it will elaborate the empirical results and provide a possible solution to the central question.
The rest parts will be structured as follows. Section Ⅱ presents the relevant literature overview and studies. Next, Section Ⅲ outlines all relevant data and methodology. Interpretation of the empirical results will be demonstrated in Section Ⅳ. Finally, Section Ⅴ provides a solution to the central question, concludes on the methodology and analysis of the experiment, and recommends some investment tips.
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II. Literature Overview and Preliminary Research
There are quantities of practical research with the intention to test or examine the effectiveness of cryptocurrency as an instrument for portfolio diversification. These researches apply different models but use similar portfolio optimization criteria. There is a lack of professional literature regarding this topic since many of those researches are not able to explain all puzzles related to this question. The most crucial difficulty involves is the estimation of risk for future prediction. This section lists some relevant literature and research, which attempt to analyze cryptocurrency and evaluate its usefulness in portfolio diversification.
2.1 Studies of Cryptocurrencies
The cryptocurrency is categorized as a digital asset that uses cryptography to secure transaction. The value of cryptocurrency is derived from Blockchain, on which digital ledgers can be shared across a network without the requirement of a central authority (Hackett 2016). Therefore, cryptocurrency is a decentralized digital asset and is highly differentiated from fiat currencies or other traditional assets.
Among the numerous cryptocurrencies, Bitcoin and Ethereum are the two most widely adopted ones in terms of market capitalization1 and unit price. Bitcoin, as the most universally accepted cryptocurrency, was created by Satoshi Nakamoto and was released in 2009 (The Economist 2015). Since its invention, quantities of altcoin variants have been derived from Bitcoin, such as Bitcoin cash. Total market capitalization accumulates to USD 107.6 billion on June 2018 from USD 1.5 billion on Apr 2013. In the meantime, the unit price soared, from USD 134 to USD 63601. However, the price of Bitcoin went through great fluctuations during a typical year or even several months in the past three years, indicating it is a speculative investment with large volatility in return. In terms of Ethereum, it is another blockchain based virtual currency. The value of Ethereum depends on the platform of Ethereum, which allows developers to leverage the application of smart contract2. Contrast to Bitcoin, Ethereum is more open-source and less restrictive. Similar to Bitcoin, the price and
1https://coinmarketcap.com/currencies/Bitcoin/historical-data/?start=20130428&end=20180814 2https://www.coindesk.com/information/ethereum-smart-contracts-work/
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aggregate market capitalization considerably increased during the past three years, along with significant price volatility as well3.
2.2 Cryptocurrencies as a Tool for Diversification
The emergence of cryptocurrency drives innovation and facilitates transactions. The questions of how to invest in cryptocurrencies and how to select portfolios with crypto assets have been continuously discussed. Initially, Bitcoin was often evaluated for the diversification purpose due to its influence and popularity in the crypto market. For instance, in Bouri et al.’s finance research, a dynamic conditional correlation model is applied to examine whether Bitcoin can be used for hedging and portfolio diversification for major world stock indices, commodity index, and some currency indices (Bouri et al. 2017). The bivariate DCC model is used to estimate the correlations between the return series of all related assets. A regression model is created to test the significance of the correlation coefficients then. According to the test results, they draw a conclusion that Bitcoin is a relatively effective tool of diversification but a poor instrument for hedging. Moreover, Bitcoin’s ability to diversify will not be constant in the long-run.
The focus of the evaluation turns to a set of cryptocurrencies instead of Bitcoin only recently. Researchers start to examine the combination of different mainstream cryptocurrencies, such as Bitcoin plus Ethereum or Bitcoin plus Litecoin. Platanakis et, al. (2018) conduct a simple research on Bitcoin, Litecoin, Ripple and Dash, aiming to examine the benefits of two different ways of diversification of those cryptocurrencies. The outcome indicates that the naïve diversification (equally weighted) is not worse than the Markowitz mean-variance optimal diversification by comparing and contrasting the Sharpe ratio and the Omega ratio. In addition, the article by Liu (2018) provides practical evidence showing that the cryptocurrency investment can be benefited from Markowitz’s portfolio theory by examining different asset allocation models. Similar to the experiment of Bouri, the examination still needs to be improved in terms of risk estimation.
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Another test is conducted by Andrianto and Diputra (2017) based on the construction of optimal portfolios. They compare the performance of a portfolio of different stock indices and different Exchange-Traded Funds (ETFs) and a new portfolio including a variety of cryptocurrencies. With the help of solver, Andrianto and Diputra form optimal portfolios by maximizing Sharpe ratios. The conclusion is that an optimal allocation of cryptocurrency between 5% and 20% can enhance the portfolio’s Sharpe ratio. The diversified portfolio significantly outperforms the S&P 500 index and the Dow Jones index. Likewise, Corbet et al.’s research (2017) also supports the effectiveness of diversification of cryptocurrencies with the finding of market isolation of cryptocurrencies from those marketable securities, particularly in short horizons.
2.3 Crypto Funds
Congruously, these researchers have found similar results and reached a consensus towards using cryptocurrency to improve portfolio diversification. In reality, there are a number of crypto funds which are actively managed by crypto managers. The composition of those funds varies according to different levels of risk appetite. Common crypto funds always consist of those mainstream cryptocurrencies, such as Bitcoin and Ethereum. A well-diversified crypto fund may include more than 20 crypto assets4. Most of the public crypto funds have achieved good performance during the last two years. The performance measures include total market capitalization, the annualized rate of return, and net investment income (deducting commissions and transaction costs). The strategy of investing in crypto funds is another indication of the significant risk involving in the cryptocurrency market. By diversified investment in a range of crypto assets, the portfolio risk will be lowered down but the portfolio return will still remain at a relatively high level. Nevertheless, crypto funds have not experienced long enough and hence the long-term performance of crypto funds lacks data evidence. It remains a doubt whether those funds can survive in the next 3-5 years, or even longer periods. Furthermore, active management always means higher commissions, which will offset the portfolio return. Yet, there is no explicit evidence proving that actively managed crypto funds can definitely beat the performance of passively managed crypto funds.
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Based on the literature review and some preliminary research, we proposed two assumptions towards the central question. The first hypothesis is that cryptocurrency does not enable portfolio diversification due to high volatility. Conversely, the second hypothesis states that the cryptocurrency is less correlated to those traditional financial assets, which advocates the usefulness of cryptocurrency in portfolio diversification. From the perspective of this paper, it is essential to conduct an experiment both to test the assumptions and verify those empirical researches. We decide to create a crypto fund with optimal weights in crypto assets allocation, rather than use a naïve diversification (equally weighted asset allocation). Then we use this optimal crypto fund to invest in existing portfolios of traditional financial assets. All portfolios are supposed to be passively managed. In addition, transaction costs of cryptocurrencies and other financial assets are not considered in the experiment.
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III. Data and Methodology
Aiming to evaluate the effectiveness of cryptocurrency as a means of diversification. we will perform the experiment examining how and to what extent cryptocurrency will work on portfolio return and risk, as mentioned in the last section. This section will interpret the relevant methods we use in conducting the experiment and all the data input into the models.
Here are some highlights of this experiment. First of all, daily closing prices (in terms of US dollars) of cryptocurrencies and the other financial assets are collected to compute their daily logarithmic returns. Secondly, 3 portfolios are constructed according to Markowitz mean-variance theory. The first one is called the crypto fund, which includes four popular cryptocurrencies. The second one is the portfolio incorporating 9 financial assets other than cryptocurrencies, which is assigned the name of the traditional portfolio. The last portfolio is a new portfolio mixing the crypto fund and the 9 traditional assets. It is called the optimal portfolio as the optimal weights of the constituent assets are achieved. Finally, a bootstrapping simulation will be performed to further evaluate the optimal portfolio with regard to its expected return and risk.
3.1 Data Collection
3.11 Number of observations
The observed period of the experiment is from 10/08/2015 to 15/06/2018, which is approximately three years. The determination of the starting date is based on the data availability of cryptocurrencies. In order to create a complete crypto fund of four cryptocurrencies with adequate population, 10/08/2015 is considered as the ideal period starting. It should be noted that almost all cryptocurrencies are tradable and marketable every second and every calendar day, whereas other financial assets are closed on weekends and public holidays. In a typical calendar year, there are 252 trading days after subtracting all bank holidays. In order to keep the consistency and eliminate biases in the calculation, daily returns of cryptocurrencies only on trading days were calculated and retained in constructing the crypto fund and the final optimal portfolio. As a result,
11 there are 719 observed dates in total.
3.12 Cryptocurrencies
We collected daily data on four mainstream cryptocurrencies, namely Bitcoin, Ethereum, Ripple and Litecoin from CoinMarketCap to construct a crypto fund. The criteria of cryptocurrency selection are total market capitalization and data availability. The market capitalization is the primary consideration because it in part reflects the intrinsic value, trading volume, and popularity among investors. Sorted by current market capitalization, the top 6 cryptocurrencies are Bitcoin, Ethereum, Ripple, Bitcoin Cash, EOS and Litecoin from highest to lowest5. Nevertheless, Bitcoin Cash is characterized as a spin-off of Bitcoin and was created in 2017. It has been publicly traded for less than one year. Likewise, EOS was developed and became public in mid-2017. The insufficiency of transaction data makes neither Bitcoin Cash nor EOS a reliable component for the crypto fund. In contrast, Litecoin is considered as a better fit than Bitcoin Cash and EOS in terms of data availability.
3.13 Other financial assets
The traditional portfolio consists of 9 traditional financial assets. With the intention of diversification, we add different stock indices, risky bonds, commodities, precious metals and private equities to this portfolio. To accurately measure the performance of these 9 traditional financial assets, we selected proximate indices for each of them.
Stock indices
The dataset of stocks incorporates the S&P 500 Index (GSPC), iShares Core MSCI Europe ETF (IEUR), iShares MSCI Emerging Markets ETF (EEM) and iShares Core MSCI Pacific ETF (IPAC). GSPC is a US stock market index and seeks to track the performance of 500 large companies, while IEUR, EEM, and IPAC seeks to track the investment performance of indices composed of large- and mid- capitalization equities in the European market, emerging market and Pacific region developed market,
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respectively. Thus, these four major stock indices can represent a considerable portion of the global stock markets. Together they can serve as the proxies for the global stock indices in the portfolio. Daily prices of GSPC is downloaded from Yahoo Finance6 while data of the other 3 stock ETF are from Investing.com7.
Commodity and precious metals
The iShares Commodity Optimized Trust (CMDT) is used as the proxy for the commodity market. It attempts to track the performance of a highly diversified investment group of commodities futures with the exposure t more than 20 different commodities, such as common metals, agricultural products, and energy. It is worth mentioning that CMDT is not a standard ETF and turns out to be speculative. However, due to the advantages of cost minimization and portfolio diversification associated with investment in commodity futures contracts, it is considered as the best representative for the performance measure of the commodity market. Daily prices of CMDT is downloaded from Investing.com. In terms of precious metals, we obtained the daily historical futures prices of gold and silver, which are the most popular choices among investors in the precious metal market. Both gold and silver are widely accepted among the investors with regard to their high intrinsic values and the role of a store of value. From a long-term perspective, they play an essential role in reducing portfolio volatility. Data for gold and silver are from Yahoo Finance.
Bonds and Private Equity
The portfolio also incorporates investing in risky bonds. The proxy for risky bonds is iShares US & Intl High Yield Corp Bond ETF (GHYG). GHYG tracks the performance of an index composed of high yield corporate bonds. The bond issuers are mainly from the developed market. Investors of GHYG pursues higher yield and desirable income and hence considered as risk-seeking. In addition, investing in private equities is also included in the portfolio. Empirical results prove that PE investing can not only provide remarkably outperformance in the long-run than marketable securities but also improve
6https://uk.finance.yahoo.com/
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portfolio diversification because it is less correlated to the market (Manigart and Wright 2011). ProShares Global Listed Private Equity ETF (PEX) is used as the proxy for private equity. PEX aims to track the performance (excluding fees) of the LPX Direct Listed Private Equity Index. Data are downloaded from Investing.com.
3.14 Risk-free rate
We chose the 10-year US treasury yield curve rate at the beginning of the investment period as the risk-free rate. The risk-free annual rate is 2.24%8.
3.2 Methodology
3.21 Daily return computation
Since all relevant data are obtained, the next step is to calculate the daily returns of all assets. Under the assumption of all financial assets will be held for at least 719 successive trading days, returns are considered as continuously compounding. As a result, we computed daily logarithmic returns for each asset, where Rlog = ln (Pt/Pt-1).
Then we derived the daily arithmetic mean return, daily variance and daily standard deviation for all the cryptocurrencies and traditional financial assets we selected. In addition, we obtained the kurtosis and skewness of daily returns in order to measure the distribution of daily returns. All descriptive statistics and interpretation will be shown in section Ⅳ.
3.22 Construction of variance-covariance matrix
The construction of variance-covariance matrices is the premise of creating an optimal crypto fund, an optimal traditional portfolio, and the final optimal portfolio. In the case of an N-asset, T-period portfolio, the variance-covariance matrix is presented on the right (Benninga 2014).
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The items on the diagonal are the variances of each asset and the remaining items are the covariances of different assets. For any asset i (0<i<N), its variance is illustrated as Var(ri) = σii. For any given σij, where i ≠ j, it refers to the covariance of the ith and the jth asset, i.e. Cov(ri, rj) = σij. The following part is going to explain the mathematic logic of computing the matrix.
First, the mean return of asset i is computed as ri = 1
𝑇 ∑ 𝑟𝑖𝑡 𝑇
𝑡=1 , i=1, …, N. Then the covariance of asset i and asset j is calculated as σij = 1
𝑇−1∑ (𝑟𝑖𝑡− 𝑟̅) ∗ (𝑟𝑖 𝑗𝑡− 𝑟̅𝑗 𝑇
𝑡=1 ), i, j
=1, …, N. Next, mean return is subtracted from the individual asset returns. A matrix of the excess returns, called E and its transpose ET are defined as follows:
E = [ 𝑟11− 𝑟̅1 ⋯ 𝑟𝑁1− 𝑟̅̅̅𝑁 ⋮ ⋱ ⋮ 𝑟1𝑇 − 𝑟̅ ⋯ 𝑟1 𝑁𝑇− 𝑟̅̅̅𝑁 ], ET = [ 𝑟11− 𝑟̅1 ⋯ 𝑟1𝑇− 𝑟̅1 ⋮ ⋱ ⋮ 𝑟𝑁1 − 𝑟̅̅̅ ⋯𝑁 𝑟𝑁𝑇− 𝑟̅̅̅𝑁 ]
The sample variance-covariance matrix is derived by multiplying ET and E and diving the product by T-1: S = (ET * E)/(T-1), which is an N x N matrix.
The creation of variance-covariance matrices can be done by using the excel array functions of MMULT () and TRANSPOSE (). There are 3 variance-covariance matrices built in total, which are a 4x4 matrix for the crypto fund, a 9x9 matrix for the traditional portfolio and a 10x10 matrix for the optimal portfolio. The matrices are built based on the daily returns. To get an annual variance-covariance matrix, we multiplied the daily matrix times 252 (trading days). The interpretation of the matrices will also be demonstrated in section Ⅳ.
3.23 The construction of portfolios
To find out the optimal portfolio, we apply the portfolio selection theory developed by Markowitz (1952). Markowitz noticed that a trade-off between expected return and risk involves in making economic investment decisions. An investor will take on additional risks only if compensated by extra returns. It is observed that on an efficient frontier, all feasible portfolios are either maximizing expected return at a given level risk or minimizing risk for a given level of expected return. Furthermore, the modern portfolio theory (MPT) introduced by Markowitz also emphasizes that a rational investor will always evaluate the risk-reward trade-off and choose a more favorable risk-reward
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portfolio (Markowitz 1991). The Sharpe ratio is probably the best risk-reward measure if returns are assumed to be normally distributed. It is defined as the excess return per unit of standard deviation (Sharpe 1994). It measures the risk-adjusted performance of an investment and provides a good indicator for portfolio evaluation. Generally, the portfolio with a higher Sharpe ratio has better a risk-reward trade-off. As a result, the basic methodology of constructing optimal portfolios is equal to seeking Sharpe ratio maximization. The return distribution is considered an important factor under the circumstance of Sharpe ratio maximization. To make our analysis sufficient, we will also examine the characteristics of return distributions of the investments.
A mathematic solution
To find out the optimal weights for each asset, we set the target to maximize the Sharpe ratio of the portfolio. The equation of the Sharpe ratio is SR = [E(rp) - rf]/σp. For an N-asset portfolio, the expected rate of return is defined as: E(rp) = ∑𝑁𝑖=1𝑤𝑖∗ 𝐸(𝑟𝑖), where wi is the weight of asset i in this portfolio, i=1, …, N. The portfolio variance is calculated as: σp2 = ∑ 𝑤
𝑖 𝑁
𝑖=1 ∑𝑁𝑗=1𝑤𝑗 Cov(ri, rj).
To maximize the Sharpe ratio SR = [E(rp) - rf]/σp, it is equal to solve the following objective function to maximize Rs:
The partial derivation of Rs in terms of the weight of asset i is as follows (Lawrence et.al 1987):
An alternative way to solve the roots of this derivative is to create vectors (Benninga 2014). Firstly, we define the excess return [E(rp) – rf] as the column
16 vector [
𝐸(𝑟1) − 𝑟𝑓 … 𝐸(𝑟𝑁) − 𝑟𝑓
]. Secondly, define a vector z, where E(rp) – rf = Sz, and S is the
variance-covariance matrix of the portfolio. Hence, z = S-1{E(rp) – rf}. Meanwhile, optimal weights are vectorized as w = {w1, w2, …, wN}, where wi = zi / ∑𝑁𝑖=1𝑧𝑖. The solution of the vector z generates the portfolio with the maximum Sharpe ratio. To simplify, the equation of optimal weights could be written as:
w = S-1{E(r
p) – rf}/Sum[S-1{E(rp) – rf}]
The outputs of the mathematic method (z-value method) are presented in Appendix 1. It should be noted that the risk-free rate we used is 2.24% per annum as stated in section 3.14 and the estimated daily risk-free rate is set as 2.24%/252 = 0.009%. Additionally, both portfolios in terms of daily returns (based on the daily matrix) and portfolios in terms of annual returns (based on the annual matrix) are constructed.
Shortcomings of the mathematic solution
According to the mathematic solution (see Appendix 1), there are negative z-values for the majority of those investments, which means that there are significant short sales involving in the optimized strategies. The math methodology allows short positions and supposes that short selling is easy to implement and the proceeds from short selling would be immediately available to investors. However, in reality, short sales are much more complicated. On one hand, part or even all of the proceeds are escrowed by brokers in the brokerage houses, which makes immediate availability quite rare (D’Avolio 2003). On the other hand, it is difficult for a typical investor to proceed on those complex and complicated short selling strategies (Surowiecki 2003). Consequently, the optimal strategy based on the mathematic solution seems to be way sophisticated for ordinary investors and turns out to be impractical in real cases. Apart from the concerns on practicality, there are some other issues should be considered if allowing short-selling, including measuring the cost of leverage, evaluating the model risk, and forecasting the fundamental risk. Therefore, short-sales will not only make the procedure technically burdensome, but also deviate the final result from its true value.
17 An alternative solution – Excel Solver
If restrictions of short selling are set, that is to say, no short sales are allowed in the investments, it will become more accessible to implement the optimal strategies. Using Excel Solver is considered as an appropriate way solving the optimal weights. The Solver parameters include the objective, which is the Sharpe ratio, dependent variables, which are the optimal weights of asset allocation we seek to, and the constraints subject to the changing variables. By setting constraints of positive values on the weights of asset allocation, optimal portfolios with maximized Sharpe ratios are generated, see
Appendix 2. Similar to the math method, both portfolios based on daily data and based
on annual data are created. It is apparent that this alternative solution makes the investment strategies more practical and more comfortable to be performed by investors. In the following part, it will focus only on the solutions of the Solver method. The results produced by the math method will not be interpreted in section Ⅳ.
3.24 Portfolio analysis
So far, we completed the construction of all optimal portfolios. Apparently, cryptocurrencies enhanced the overall performance with a remarkable increase in portfolio return and the Sharpe ratio. Nevertheless, portfolio volatility increased significantly in the meantime. It is plausible to draw a conclusion that cryptocurrencies can play an active role in portfolio diversification and facilitate higher yields. However, it raises a question whether the performance would remain stable and persistent in the future. It is actually crucial to deal with the estimation of portfolio risk in portfolio management, especially for investments which are less predictable and highly volatile such as the cryptocurrencies. Therefore, a relevant procedure in terms of performance prediction should be conducted to enhance the robustness of our experiment.
3.25 Bootstrapping simulation
Bootstrapping is in a sense a statistic test involving resampling procedure, initially introduced by Efron (1979). It is defined as a computation-intensive method for estimating the distribution of a parameter, such as mean and variance, by resampling the data. In brief, a basic bootstrap method extracts N samples from a population/sample with random replacement from that population/sample. Note that every sample could
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be selected more than once. The new dataset after resampling is used as the inputs for further estimation. The research of Delcourt and Petitjean (2011) has provided empirical results of the resampling efficiency on portfolio optimization. The strategy of resampled optimization results in a portfolio with more stability and a higher degree of diversification than a traditional mean-variance efficient portfolio. Simplicity is one of its great advantages of the bootstrap method. It offers a straightforward method to derive estimates of confidence intervals and standard errors for complex parameters. Moreover, bootstrapping also provides a higher degree of reliability for ascertaining the stability of statistic results. (Efron and Tibishirani 1993). The bootstrapping simulation has been widely applied in testing the efficiency of portfolios, particularly more appealing to multi-factor models with limited sample size. For instance, Chou and Zhou’s research (2006) provides arguments that bootstrapping simulation does improve the reliability of testing efficiency in the examination of an optimal portfolio of CRSP value-weighted stock indices, regardless the assumption of distribution of stock returns. According to our findings (see section 4.12 and Table 5), the returns of all assets we evaluate in this paper are hardly normally distributed. Hence, bootstrapping could be the best way to detect the stability of empirical results further.
Scherer (2002) summarized the common methodology of bootstrap resampling for portfolio optimization, which is outlined as follows:
1. Build an estimated variance-covariance matrix and mean return vectors based on historical data.
2. Resample from historical data by extracting N draws from input distribution and create a new variance-covariance matrix based on samples randomly selected.
3. Compute the efficient frontier from inputs derived in step 2 and save the optimal portfolio weights.
4. Find out the average optimal weights for each asset in the portfolio by repeating step 2 and step 3 M rounds.
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In our case, the sample size we randomly selected from the population is 252, which means the performance of the portfolio in 252 trading days will be simulated. In addition, we set the number of simulations to 20 rounds. Therefore, we have N=252 and M=20. This simulation is performed via Excel VBA9. The bootstrapping procedure that we apply in predicting the future performance is outlined as follows:
1). We generate a new table of (1+ historical daily return) for each asset, in order to get the geometric mean return. Based on the new table, we estimate the expected annual return (using the GEOMEAN function) and a variance-covariance matrix of all financial assets (9 traditional assets plus the crypto fund).
2). We assign a random number between 0 and 1 (using RAND function) to each observed sample and randomly sort out 252 samples from the population. After resampling, we calculate a new variance-covariance matrix and new geometric mean return based on the new samples. The randomly selected samples are meant to construct a new portfolio. Next, we use functions to calculate the return, variance (standard deviation) and the Sharp ratio of the new portfolio. In the meantime, we set a constraint on the new portfolio of Sharpe ratio maximization. The optimal weights satisfying the constraint can be derived and saved after one round.
3). After repeating the above step 20 rounds, which means 20 new optimized portfolios are created, we take the average return, variance-covariance matrix, as well as the average bootstrapped weights of each asset. Thus, it will be easy to derive the portfolio return and volatility, along with the portfolio Sharp ratio.
In fact, in order to increase the accuracy of prediction, more than one bootstrapped portfolio could be generated by conducting the procedures 1) to 3) several times. The bootstrapped portfolio weights and variance-covariance matrix are illustrated in
Appendix 4.1 and Appendix 4.2 respectively. Section 4.4 will elaborate on the
simulated portfolio.
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IV. Empirical Results 4.1 Descriptive Statistics
The descriptive statistics provide an intuitive preview of characteristics of the return distribution. We calculated the daily return, daily risk and kurtosis and skewness of daily returns based on the daily data we obtained. The daily return and variance are projected to annual data by multiplying 252 (days). The same method applies in generating the annual variance-covariance matrices. The construction of the optimal portfolios is based on the annualized data. Mean return and standard deviation are the straightforward indicators of the performance of an investment, whereas skewness and kurtosis measure the distribution of return. Skewness measures the extent to which the return distribution is asymmetric to the mean value of return (Brooks 2014). Meanwhile, kurtosis defines the fatness of the distribution tails and measures how the return series is peaked at the mean value. For a normally distributed series, the coefficient of skewness equals zero and the coefficient of kurtosis equals 3 (Ditto).
4.11 Cryptocurrencies
Table 1 shows the daily return, risk, and return distribution of all cryptocurrencies in
the crypto fund. It is clear that each cryptocurrency is able to yield an impressive daily return but the daily volatility of each cryptocurrency is also high. Projected to annual data, i.e. Table 2, all of them can generate more than 100% rate of return with considerably high variance and standard deviation. The higher the rate of return, the more volatile the crypto asset is. Compared with Ethereum, Ripple, and Litecoin, Bitcoin is the least volatile one despite the strong instability showed in its price in the past 6 months. In terms of distribution of daily return, it is noticeable that those four cryptocurrencies show strong fat tails in the distribution of return with significantly positive values (greater than 3) of kurtosis. The heavy-tailed distribution indicates that
Bitcoin Ethereum Ripple Litecoin
Mean return 0.44% 0.91% 0.57% 0.45%
Variance 0.23% 0.77% 0.76% 0.49%
Standard Deviation 4.76% 8.78% 8.72% 7.00%
Kurtosis 4.68 5.17 16.94 13.30
Skewness -0.17 0.82 2.17 1.69 Table 1: Daily descriptive statistics of cryptocurrencies.
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Bitcoin Ethereum Ripple Litecoin
Mean return 111.90% 229.58% 144.11% 112.52%
Variance 57.04% 194.44% 191.81% 123.53%
Standard Deviation 75.53% 139.44% 138.50% 111.14% Table 2: Annual descriptive statistics of the cryptocurrencies.
the investors holding crypto assets will more likely to experience extreme returns while the extreme returns could be either positive or negative. Higher kurtosis thus means a higher risk associated with the investment. On the other hand, skewness coefficients of the cryptocurrencies are different from zero, meaning a lack of symmetry of the return distribution. The negative skewness of Bitcoin means that the distribution is left-skewed and implies that there are more numbers of above-average return than below-average return. On the contrary, the positive skewness of the other three cryptocurrencies means that the distributions are right-skewed and indicates that the number of below-average return exceeds the number of above-average return (NIST 2013). The risks associated with skewness will bring about difficulties in future prediction, because the more the skewness deviates from zero, the less the returns are normally distributed and less predictable.
Table 3 illustrates the variances of each cryptocurrency and the covariance of different
cryptocurrencies on annual basis. In order to have a more intuitional perspective, we generate a matrix of correlations between those cryptocurrencies, see Table 4.
Bitcoin Ethereum Ripple Litecoin
Bitcoin 57.04% 35.75% 31.65% 45.45%
Ethereum 35.75% 194.44% 32.06% 48.06%
Ripple 31.65% 32.06% 191.81% 50.46%
Litecoin 45.45% 48.06% 50.46% 123.53%
Table 3: Var-Cov matrix of the cryptocurrencies on annual basis.
Bitcoin Ethereum Ripple Litecoin
Bitcoin 1.00
Ethereum 0.34 1.00
Ripple 0.30 0.17 1.00
Litecoin 0.54 0.31 0.33 1.00 Table 4: Correlation matrix of the cryptocurrencies
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The correlation coefficient is a measure of the degree to which two different assets move together. The value of the correlation coefficient ranges from -1 to 1. According to Bodie et. al (2011), a correlation coefficient with the value of 1 means the returns of two assets change in the same direction while a value of -1 means the returns move in an opposite way. If the value is between 0.5 and 1, this means a strong and positive relationship between the two assets. In addition, if the value ranges from 0.1 to 0.3 and 0.3 to 0.5, it indicates the two assets are weak-positively and moderate-positively correlated respectively. Based on the correlation matrix, the four cryptocurrencies are either weakly correlated or moderately correlated, with the only exception of the pair of Bitcoin and Litecoin. Even though the individual risks of the cryptocurrencies remain at a relatively high level, it is considered as a good idea to diversely invest in a variety of crypto assets due to the low correlations between them. This also partly explains the reason why we invest in a crypto fund instead of an individual cryptocurrency.
4.12 The crypto fund and other financial assets
Table 5: Daily descriptive statistics of all financial assets.
The daily descriptive statistics are presented in Table 5. At this stage, the crypto assets are considered as an integer after the construction of the optimal crypto portfolio. Obviously, the crypto fund significantly outperformed the other 9 financial assets during the whole observed period. The daily mean return is 15 times larger than the S&P 500 index and 31 times larger than either gold or silver. Certainly, the daily variance and standard deviation of the crypto fund are also significantly larger than
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other traditional financial instruments. Again, projected to annualized data in Table 6, the annual return of the crypto fund is greater than 160%, which is a very attractive investment for those investors who are speculative and seeks exceptional rate of return. At the first glance of Table 6, only the investment in the private equity ETF yields a negative return for the past three years. Not only is the mean return negative, but also great risks are associated with investing in PEX. It seems that such kind of investments should be out of the scope of an optimal portfolio. Furthermore, the optimal portfolio ought to eliminate some other financial instruments which are less likely to generate a desirable risk-adjusted return, namely the financial assets with relatively low Sharpe ratios. For example, the investments in IEUR, CMDT, and silver are unlikely to be efficient and hence should be taken out of the consideration. The result could either be an appropriate reflection of the truth, or just an unusual scene in a short time window. Further analysis needs to be done to verify the validity of the result.
In terms of the distribution of return, the crypto fund as a whole turns out to be less skewed and less asymmetric compared with individual cryptocurrencies. The kurtosis of 3.62 and the skewness of -0.28 indicates the return is quite close to being normally distributed. The feature of the normal distribution of return will absolutely enhance the accuracy and robustness in risk estimation by using financial models. Moreover, it makes more sense to create a crypto fund rather than investing in a single cryptocurrency like Bitcoin. Unlike crypto assets, the return distribution these traditional financial assets tend to be less skewed and more symmetric based on their kurtosis and skewness in Table 5, except IEUR and PEX. The extremely large values of kurtosis of IEUR and PEX (even greater than individual cryptocurrencies), as well as the skewness of their return distribution, provide an alternative explanation of their inappropriateness as feasible investments. For the rest of the traditional financial assets, the return distributions are seemingly feasible for future prediction. Overall, intuitively, the investment of S&P 500 index, EEM index, IPAC index and the gold, are pondered as reliable and appropriate compositions for the construction of an optimal portfolio with regards to the reward-to-volatility trade-off and the return distribution. It will be discussed in section 4.3.
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4.2 Correlations between Cryptocurrencies and Other Financial Assets
The daily and annual variance-covariance matrices of the crypto fund and those traditional financial assets are built not only to examine the relationships between different assets, but also to derive the optimal weights for each cryptocurrency. The matrices are demonstrated in Appendix 3. Focus on the annual matrix, the crypto fund per se has the highest variance amongst all the financial assets. Nevertheless, the covariance of the crypto fund and the traditional assets are not material and even slightly negative. This finding is in line with our assumption and research. On the contrary, the variances between those marketable securities, namely S&P 500, IEUR, EEM, IPAC, as well as marketed private equities, are positively significant. However, for commodities and precious metals, they have lower covariances with other assets. Gold, in particular, has negatively insignificant covariances with all marketable securities. In order to make the relationships among the assets unambiguous, we generate a matrix of correlation coefficients of all financial instruments, as illustrated in Table 7.
Assets
S&P
500 IEUR EEM IPAC CMDT Gold Silver GHYG PEX Crypto fund S&P 500 1.00 IEUR 0.76 1.00 EEM 0.79 0.78 1.00 IPAC 0.79 0.76 0.80 1.00 CMDT 0.27 0.30 0.35 0.30 1.00 Gold -0.16 -0.12 -0.01 -0.08 0.12 1.00 Silver 0.07 0.15 0.22 0.15 0.25 0.74 1.00 GHYG 0.47 0.54 0.51 0.47 0.25 0.00 0.12 1.00 PEX 0.42 0.52 0.45 0.48 0.22 -0.08 0.09 0.28 1.00 Crypto fund 0.06 0.03 0.04 0.04 -0.01 0.06 0.05 0.06 -0.07 1.00
Table 7: Correlation matrix of all financial assets
According to Table 7, the correlation coefficients between the crypto fund and other traditional assets are within the range of -0.1 and 0.1. Based on the criteria provided by Bodie et.al (2011), there is no linear relationship between the crypto fund and other financial instruments. The immaterial correlation implies that the crypto assets can serve as an appropriate vehicle for diversification. Similarly, gold, as a value-preserving investment with high economic value, has negatively weak relationships with marketable securities and moderately positive relationship with the remaining
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traditional instruments. Thus, it is also pondered as a feasible diversification tool. In terms of the stock indices, they are highly correlated with each other, indicating that they as a whole might not well exert a desirable influence on portfolio diversification. Additionally, the risky bond GHYG and private equity fund PEX may not play an effective role in portfolio diversification either, as they are strongly or moderately correlated with the stock indices. So far, the study on the examination of the correlations between cryptocurrencies and traditional assets confirms that there is no linear relationship between them. One of the two assumptions has been verified with a satisfying answer. Yet, the influence that the cryptocurrencies exerting on the portfolio still remains a doubt. The potential influence is not only on the current portfolio performance but also on the future stability and reliability.
4.3 Evaluation of the Optimal Portfolio
With the intention of examining the change in portfolio performance, it is necessary to contrast the performance of the traditional portfolio and the final optimal portfolio. By using the Excel solver, the optimal weights for all investments have been found. Table
8 shows the results of the optimized traditional asset portfolio, while Table 9 shows the
results of the optimized portfolio consisting of both traditional assets and crypto assets.
The optimal traditional portfolio suggests that a normal investor can achieve Sharpe ratio maximization with the strategy of investing in S&P 500 index and gold only. This is somewhat corresponding to the analysis in the last section. We have obtained some findings that not all of the investments in the traditional portfolio are efficient and reliable. They are either unable to generate good excess returns (judged by their insignificant Sharpe ratios) or highly volatile in the past and the future. The S&P 500 index, as the best indicator of the US stock market, tracking 500 US-listed companies with leading performance and large market capitalization, is undoubtedly the better, if not the perfect, stock investment among the security investments. By conducting a naïve verification, which is to calculate the individual Sharpe ratios of each asset (see
Table 10), we found that S&P 500 index has the second highest Sharpe ratio (0.60)
following the crypto fund (1.96), whereas the Sharpe ratios of the other traditional assets are insignificant or even neglectable.
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Table 8 Table 9
Note: The returns and risk measurements are annualized.
The investment of CMDT, GHYG, and PEX even yields negative individual Sharpe ratios. In terms of gold, it might not be competitive in improving the portfolio return, while it can be used as one of the best diversification tools due to its market isolation. The rest of the traditional assets could not be able to achieve either of the purposes. They can be feasible for some advanced and sophisticated investors when complicated and instant short selling strategies are allowed and possible, with the reference to
Appendix 1. As explained in section 3.23, those scenarios are rare in real-world cases.
Finally, we are suggested to invest 66.5% of total capital allocation in S&P 500 and 33.5% in gold. The optimal traditional portfolio yields an annual mean return of 8.58% and annual volatility of 9.26%, with the maximized Sharpe ratio of 0.68.
Investments
S&P
500 IEUR EEM IPAC CMDT Gold Silver GHYG PEX Cryptofund
Sharpe ratio 0.60 0.03 0.28 0.23 -0.09 0.23 0.07 -0.19 -0.38 1.96 Table 10: The individual Sharpe ratios of each asset (on annual basis).
Regarding the optimal portfolio including crypto investments (see Table 9), an investor is recommended to invest in S&P 500 index, gold, as well as the crypto fund, with the proportions of 50%, 19.4%, and 30.6% respectively. It does not suggest any investments of the other traditional financial assets either. An allocation of 30.6% in the crypto assets facilitates the maximum Sharpe ratio of this portfolio. The portfolio
Investments Weight S&P 500 66.50% IEUR 0.00% EEM 0.00% IPAC 0.00% CMDT 0.00% Gold 33.50% Silver 0.00% GHYG 0.00% PEX 0.00% Sum 100% Mean return 8.58% Variance 0.86% Portfolio Volatility 9.26% Sharpe Ratio 0.68 Investments Weight S&P 500 50.01% IEUR 0.00% EEM 0.00% IPAC 0.00% CMDT 0.00% Gold 19.40% Silver 0.00% GHYG 0.00% PEX 0.00% Cryptofund 30.59% Sum 100.00% Portfolio Return 55.16% Variance 6.79% Portfolio Volatility 26.05% Sharpe Ratio 2.03
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volatility might not be minimized while it acquires the maximum excessive return given the annual risk-free rate of 2.24%. The annual mean return of the optimal portfolio rises to 55.16%, increased by 543%. Moreover, the portfolio Sharpe ratio is also remarkably enhanced by nearly 200%, becoming 2.03. Meanwhile, the variance and standard deviation of the portfolio are also considerably enlarged compared with the traditional portfolio. Overall, the optimal portfolio with investments in cryptocurrencies achieves outperforming a portfolio without cryptocurrencies. At this stage, we are more confident to ascertain the usefulness of the cryptocurrencies in portfolio diversification. However, it is not the end of the question because there still exists a puzzle in relation to the expected return and estimated risk in the predictable future.
4.4 A Prediction of the Expected Return and Risk
Table 11 presented below gives an overview of a bootstrapped portfolio after
conducting 20 rounds of bootstrapping simulation.
Investments Expected annual return forecasts Bootstrapped weights 1 (GSPC) 7.01% 12.03% 2 (IEUR) -0.98% 0.37% 3 (EEM) 6.57% 2.95% 4 (IPAC) 2.31% 4.48% 5 (CMDT) 4.41% 22.71% 6 (GOLD) 5.92% 14.86% 7 (SILVER) 5.32% 2.94% 8 (GHYG) 0.90% 4.42% 9 (PEX) -5.12% 6.50% 10 (Cryptofund) 235.62% 28.74% Sum 100.00% Exp.port.return 70.60% Exp.port. SD 24.08% Sharpe ratio 2.84
Table 11: Bootstrapped optimal portfolio
Compared to the mean-variance optimized portfolio (as presented in Table 9), the simulated portfolio leads to a higher return with an even lower standard deviation. Moreover, the Sharpe ratio has increased by 40% and reaches 2.84. The abnormal return could be caused by the short time window, again. It is hard to predict such significant rate of return would occur a second time in the future of another three years. Another noteworthy difference between the two optimal portfolios is that the bootstrapped
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portfolio is more diversified than the mean-variance optimized portfolio. It suggests investing in all of the financial instruments despite quite small proportions in some of them. The larger degree of diversification is caused by the features of the resampling methodology. Randomly sorting out 252 samples from a pool 719 observations will inevitably lead to much cross-correlation due to high chance of one sample being repeatedly selected. It has been argued by Boris and Aleksandra (2012) that all resampled portfolios will suffer from the deviation of the parameters because the true distribution of the input is uncertain. This is also the biggest disadvantage of bootstrapping resampling. However, looking into the weights of 20 bootstrapped portfolios in Appendix 4.1, a large quantity of the capital is allocated in S&P 500 index, the commodity ETF, gold and the crypto fund. Particularly, almost all of the 20 rounds suggest the investment of cryptocurrencies. The usual outcome is caused by some extreme cases. In addition, by contrasting the variance-covariance matrices generated by the mean-variance optimization method and the bootstrap method (Appendix 3.2 versus Appendix 4.2), there are no noticeable differences, indicating the portfolio risk will not significantly fluctuate. To sum up, the results of the bootstrapping simulation lead to more stable and more diversified portfolios. This satisfying outcome implies the performance of the optimal portfolio will be stable (or even better) and reliable in the foreseeable future. Actually, we have performed more times of simulation. The outcomes do not considerably vary each other so the rest of the bootstrapping results are not presented or explained here.
4.5 Solutions to the Central Question
All relevant procedures of the experiment have been completed thus far. The empirical results have helped answer the central question raised at the beginning. At the current stage, we can hold an opinion that cryptocurrency can indeed serve as a useful tool for portfolio diversification to a large extent. On one hand, our findings show that cryptocurrencies are hardly correlated with other financial instruments and are highly likely to be safe facing systematic risks. The relative isolation of crypto assets makes them expose less to market shocks. On the other hand, our experiment offers sufficient proof of the improvements in portfolio performance if we add cryptocurrencies to traditional financial assets, even if cryptocurrencies themselves are risky assets.
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Through a reasonable diversification, a crypto fund with optimally diversified cryptocurrencies does significantly enhance the ability to achieve outstanding excess returns. Last but not least, the estimated return and risk is predicted to be stable based on the results of the bootstrapping simulation.
According to the mean-variance optimal weights and the bootstrapped optimal weights, the optimal allocation of cryptocurrency in a portfolio should be around 30%. In turn, we can derive the optimal weights of each crypto assets in the optimal crypto fund (see
Appendix 2.1). The optimal weights for Bitcoin, Ethereum, and Ripple are 14.1%,
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V. Conclusion
It leads to the conclusion that cryptocurrency can serve as an effective tool for portfolio diversification to a large extent. Firstly, the correlation of the crypto fund and other asset classes shows there is no linear relationship between the crypto assets and the traditional financial instruments. The cryptocurrency is not as vulnerable as the traditional financial assets especially the stock indices when the market suffers from shocks. In fact, the cryptocurrency market is relatively isolated from the financial market. As a result, the combination of crypto assets and financial assets benefits the effect of diversification. Secondly, the optimal portfolio including both crypto assets and traditional assets significantly outperforms the traditional portfolio with 200% increase in the Sharpe ratio. Despite lower rate of return compared to the crypto fund, the optimal portfolio also beats the crypto fund in terms of the Sharpe ratio. This result not only supports the perspective of cryptocurrency for diversification, but also elucidates the importance of portfolio diversification. It could be regarded as kindly advice for investors who only invest in cryptocurrencies. It should be noted that exceptional returns of cryptocurrencies come along with considerable amount of volatility. The risk-reward trade-off is not maximized and therefore the crypto fund is not as efficient as the optimal portfolio. Finally, the bootstrapping simulation also indicates the effect of cryptocurrency influencing on portfolio diversification would be stable. The estimated risks are basically consistent to what the optimal portfolio produces. The bootstrap method will lead to a more diversified portfolio due to the resampling. However, cryptocurrencies, as well as S&P 500 and gold, are still favorable investments. It is basically consistent with the suggestion of the optimal portfolio.
However, there are limitations existing in our models and methodology, which means further study and research need to be continued. First of all, the selection of observation period is on the purpose of data sufficiency of the cryptocurrencies. However, it leads to a lack of sample size. The three-year short window makes the dataset of daily return too small to be accurate. This results in some unusual return patterns, which might indicate inappropriate investment strategies. Another limitation lies in constructing the Sharpe ratio maximization portfolios. A premise of maximizing the Sharpe ratio is returns are assumed to be normal. However, not all assets have normal- or
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approximately normal return distributions. It is very difficult to totally eliminate the negative effects of this drawback in conducting our experiment. The final optimal weights could be biased and the true optimal investment strategy could be deviated.
To summarize, there are three recommendations for investors who seeks to invest in cryptocurrencies and who have already actively participated in the crypto market. First, be clear of your level of risk appetite. Some investors prefer to minimize risk, whereas others would like to seek risk-return trade-off maximization. The portfolio including cryptocurrencies tends to generate greater excess return, while the portfolio risk increases as well. This would be unfavorable for investors who are risk averse. Second, a well-diversified portfolio is more likely to achieve better performance. The optimal portfolio is the example. The overall performance is enhanced of the optimal portfolio compared with either the crypto fund or the traditional portfolio. Last but not least, the optimal weights of cryptocurrency invested in the optimal portfolio is approximately 30%, which is a rough estimation. Ideally, the investor will allocate the rest of the capital to S&P 500, gold, and perhaps a bit in commodities.
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Appendices
Appendix 1 - Optimal weights using the mathematic solution
1.1 Optimal weights of the crypto fund
Crypto fund-Daily Return
Cryptocurrencies Z-Value Weight Bitcoin 1.19 49% Ethereum 0.90 37% Ripple 0.41 17% Litecoin -0.06 -2% Sum 2.44 100% Portfolio Return 0.638% Portfolio Variance 0.26% Portfolio Volatility 5.09% Sharpe Ratio 0.1242
1.2 Optimal weights of the traditional portfolio
Portfolio Daily Portfolio Annual
Financial Instruments Z-Value Weight Financial Instruments Z-Value Weight S&P 500 12.08 343% S&P 500 11.61 1081% IEUR -2.97 -84% IEUR -2.87 -267% EEM 0.19 5% EEM 0.59 55% IPAC -0.67 -19% IPAC -0.80 -75% CMDT -0.87 -25% CMDT -1.24 -115% Gold 4.86 138% Gold 4.17 388% Silver -1.27 -36% Silver -1.06 -99% GHYG -4.92 -139% GHYG -6.30 -587% PEX -2.90 -82% PEX -3.03 -282% Sum 3.53 100% Sum 1.07 100%
Mean return 0.159% Mean return 126.18%
Variance 0.043% Variance 115.44%
Portfolio Volatility 2.085% Portfolio Volatility 107.44%
Sharpe Ratio 0.07 Sharpe Ratio 1.15 Crypto fund-Annual Return
Cryptocurrencies Z-Value Weight Bitcoin 1.18 48% Ethereum 0.90 37% Ripple 0.41 17% Litecoin -0.06 -2% Sum 2.43 100% Portfolio Return 160.90% Portfolio Variance 65.32% Portfolio Volatility 80.82% Sharpe Ratio 1.96
36 1.3 Optimal weights of the new portfolio
Portfolio Daily
Investments Z-Value Weight
S&P 500 10.60 292% IEUR -2.98 -82% EEM 0.48 13% IPAC -0.93 -26% CMDT -0.27 -7% Gold 4.01 111% Silver -1.39 -38% GHYG -6.53 -180% PEX -1.72 -47% Cryptofund 2.35 65% Sum 3.63 100% Mean return 0.54% Variance 0.15% Sigma 3.84% Sharpe Ratio 0.139
Appendix 2 - Optimal weights using Solver
2.1 Optimal weights of the crypto fund
Cryptofund-Daily
Cryptocurrencies Z-Value Weight
Bitcoin 1.16 47% Ethereum 0.90 37% Ripple 0.41 17% Litecoin 0.00 0% Sum 2.47 100% Portfolio Return 0.636% Portfolio Variance 0.26% Portfolio Volatility 5.07% Sharpe Ratio 0.12 Portfolio Annual
Investments Z-Value Weight
S&P 500 10.12 861% IEUR -2.87 -244% EEM 0.89 75% IPAC -1.06 -90% CMDT -0.64 -54% Gold 3.33 283% Silver -1.18 -100% GHYG -7.92 -674% PEX -1.84 -157% Cryptofund 2.35 200% Sum 1.18 100% Mean return 414.88% Variance 351.03% Sigma 187.36% Sharpe Ratio 2.202 Cryptofund-Annual
Cryptocurrencies Z-Value Weight
Bitcoin 1.16 47% Ethereum 0.91 37% Ripple 0.41 17% Litecoin 0.00 0% Sum 2.47 100% Portfolio Return 160.40% Portfolio Variance 64.95% Portfolio Volatility 80.59% Sharpe Ratio 1.96
