The Benefits of Adding Gold and Bitcoin to an Investment Portfolio: From
the Perspective of a European Investor.
-R. van der Goot
Master’s Thesis Econometrics, Operations Research and Actuarial Studies.
-Supervisor: dr. E.L. Kramer
The Benefits of Adding Gold and Bitcoin to an Investment
Portfolio: From the Perspective of a European Investor.
R. van der Goot June 22, 2020
Abstract
Contents
1 Introduction 5
2 Methodology 10
2.1 Univariate GARCH Models . . . 10
2.2 Behaviour of the Tail . . . 11
2.3 Dependence and Copula Models . . . 12
2.4 Risk Measures . . . 14
2.5 Procedure . . . 15
3 Data Analysis 17 3.1 Preliminary Analysis . . . 17
4 Results 24 4.1 Portfolios Including Gold . . . 29
4.2 Portfolios Including Bitcoin . . . 31
4.3 Portfolios Including Gold and Bitcoin . . . 33
4.4 Economic Recovery Scenarios . . . 36
4.5 Sensitivity Analysis . . . 39
5 Discussion 41
6 Conclusion 44
1
Introduction
Ray Dalio is the CEO of Bridgewater associates, the largest hedge fund in the world. In his new book Dalio (2020) talks about a long-term debt cycle repeating itself throughout history. In this debt cycle, lasting about 50 to 75 years, we start of in a situation of ’hard money’, in which money is directly linked to an asset of intrinsic value like gold and low amounts of debt. However, slowly during this cycle we see a deterioration of the ’hard money’ into a system based on faith and high amounts of debt. Currently, we are facing the ending of a cycle which began after world war two, when the US dollar became the reserve currency. Throughout history, governments and central banks have always reduced their debt burden by either; printing excessive amounts of money to cause inflation, by devaluing their currency, or by defaulting on their debt. Examples of this can be traced back to the old testament, with mention of the year of Jubilee, occurring every 50 years, in which all debts were forgiven (Leviticus 25:8-13). More recently, during the ending of the last cycle, Franklin D. Roosevelt changed the amount of dollars per ounce of gold from 20.67$ to 35$ causing an enormous decrease in the purchasing power of investors paper money. Therefore, Dalio (2020) states that holding debt is a ticking time bomb that is rewarding early in the cycle and deteriorates in risk to reward until a big default or devaluation occurs. Moreover, as investors only experience one of these cycles in their lifetime, they will not anticipate the ending of a cycle, which will likely result in large losses for their portfolios.
Whether you believe Dalio’s theory about the big debt cycle, it is obvious that holding debt has been becoming less attractive. The Yield on AAA (triple A) government bonds is negative, and investors can only obtain a positive yield when they take on the additional risk. It is furthermore unlikely that interest rates decrease much further, which in the past has led to high returns on long-term bonds. When we also take into account inflation, the real return on bonds can easily reach up to -2% per year. Therefore, holding these types of bonds is slowly eroding purchasing power even without considering the risk of currency devaluations and defaults. In the past, stocks have performed strongly in periods of high inflation. However, currently central banks are running out of options to stimulate the real economy. Therefore, the coming downturn can result in both lower economic output and higher inflation. This will be detrimental for a portfolio mainly consisting of stocks and bonds.
For the reasons stated above, this thesis tries to investigate whether adding gold or Bitcoin to a portfolio consisting of stocks and bonds, adds diversification benefits or improves the risk-to-reward trade-off of an investment portfolio. We included gold because it has a great history of insuring purchasing power and has grown by the rate of inflation on average (Ghosh et al. 2004). The main problem with holding gold is that it does not give an interest or dividend. However, currently, bonds yield negative interest and have the risks of default and inflation. Gold is immune to these risks because holding gold does not require another party and gold is a great hedge against inflation. Many influential players agree with this point. For example, investment group CLSA stated that they recommend pension funds to hold 50% in physical gold, 20% in gold mining stocks and 30% in Asia ex Japan stocks (Barrons 2016). Besides that, Ray Dalio the CEO of Bridgewater Associates also states that in the new environment, where central banks want to devalue their currency, adding gold to a portfolio is both return enhancing and risk reducing.
central authority can create additional Bitcoins like in the fiat money system. Due to its similar-ities with gold, the proponents of Bitcoin have dubbed it digital gold (Popper, 2015). They also argue that Bitcoin is a great store of value and functions as a safe haven asset. However, there are some obvious differences between gold and Bitcoin. Firstly, gold is a proven safe haven asset and currency throughout history, whereas Bitcoin is currently mainly used for speculation and has not proven itself as such. Besides that, gold has obvious intrinsic value and is used in the real economy, whereas Bitcoin has no intrinsic value and is only valuable as long as people perceive it to be. Lastly, the market for gold is one of the most liquid markets in the world. Gold is traded almost 24 hours a day and gold is used for many applications like monetary policy, jewellery and even religiously in countries like India. Contrary to this, the market for Bitcoin is a notoriously illiquid market. We argue that in an uncertain environment, gold is a stable investment, and are curious to analyse the new asset class of Bitcoin, since its main purpose is to be a great store of value. Therefore, we will investigate whether adding either gold or Bitcoin to a benchmark portfolio consisting of stocks and bonds gives diversification benefits or improves the risk to reward trade off. Currently, the literature on the benefits of adding gold and Bitcoin to a portfolio focusses mainly on historical returns. For example, Jaffe (1989) found that including gold to a portfolio from Sept-1971 to June-1987 improved the risk-to-reward trade-off. However, later research by Johnson Soenen (1997) found that including gold to a portfolio from 1984 to 1995 did not give any significant improvements. This is further investigated in the literature review section, however, this already shows that since returns and correlations have greatly differed per period, it is clear that the results of these papers are greatly dependent on the time period we choose. Therefore, instead of using historically calculated expected returns, this thesis makes use of forward looking expected return assumptions. Using these expected return assumptions, we focus on the benefits of adding gold and Bitcoin to a portfolio for the future period instead of a given historical period. To do this, we fit our model on historical returns and simulate 5000 new future return series based on this model. We then transform the simulated returns in such a way that we keep the inherent volatility and correlations, however, change the expected returns to our future expected return assumptions. We think this gives our predictions more validity for the coming period, because it is unrealistic to assume that Bitcoin will double in price every year like it did in previous years. This thesis therefore tries to add to the literature by investigating the benefits of adding gold and Bitcoin to a portfolio for a future period instead of older literature that only has a backward looking approach. A forward looking approach is also more relevant for investors to improve their asset allocation for the coming period. Besides that, we will investigate different correlation structures by looking at two different time periods. One of the time periods is including the corona crisis and the other is excluding it. We are now ready to state the main research question we want to answer: Are gold and Bitcoin appropriate diversifiers in a portfolio consisting of stocks and bonds for a European investor, and do gold and Bitcoin improve the risk-to-reward trade-off in a European investors’ portfolio consisting of stocks and bonds? To answer this question we want to investigate the seven sub-questions given below:
1. Are gold and Bitcoin uncorrelated to stocks and bonds?
2. Are gold and Bitcoin safe haven assets during economic uncertainty?
3. Does adding gold or Bitcoin to a portfolio consisting of stocks and bonds reduce the risk of the minimum risk portfolio?
5. What is the optimal allocation of gold and Bitcoin in an investment portfolio consisting of stocks, bonds, gold and Bitcoin?
6. Is this optimal allocation of gold or Bitcoin dependent of the future economic environment? 7. Do gold and Bitcoin have the same investment characteristics. i.e. is Bitcoin the ’digital
gold’ ?
To investigate the first two questions we look at the Pearson correlation, Kendal’s Tau rank correlation and the coefficient of tail dependence between gold, Bitcoin, stocks and bonds. To answer questions three to five, we optimize a portfolio consisting of stocks, bonds, and gold and/or Bitcoin and compare this portfolio with a benchmark portfolio consisting of stocks and bonds. We compare the risk of the portfolios based on the Mean-Variance and the 2.5% conditional value at risk (CVaR). To answer question three, we examine whether the risk of the optimal portfolio is reduced without requiring a minimum return. For question four, we examine the risk reducing properties of gold and Bitcoin while requiring minimum returns of 0, 1, 2 and 3% on our portfolio, which will show us whether the risk-to-reward trade-off is improved. The results obtained for questions three and four will also result in an optimal weight of gold and Bitcoin that minimizes the risk of our portfolio, while giving us a certain minimum return, thereby answering question five. To answer question six, we will be looking at four plausible future economic recoveries out of the COVID crisis. For each scenario, we will examine the optimal weight of gold and the risk reducing properties of gold and Bitcoin, assuming this scenario plays out. Lastly, we answer question seven by comparing the results of questions one to six for both gold and Bitcoin. This will give us an answer to whether the assets are similar in investment characteristics.
Literature Review
GoldSince 1971, there have been many papers on the advantages of gold in an investment portfolio. Most of this research investigates the hedging, diversification and safe haven properties of gold. Baur and Lucey (2010) define a hedge as an asset that is negatively correlated or uncorrelated to a portfolio on average. They define a diversifier as an asset that is positively correlated, however not perfectly correlated to a portfolio on average. Lastly, they define a safe haven asset as an asset that is negatively correlated or uncorrelated to a portfolio during periods of market uncertainty. However, many other definitions are used throughout the economic literature. In this thesis we will also be using a different definition for a diversifier. This is because previous research by Symitsi et al (2019) showed that, although Bitcoin has a low correlation with traditional assets, it does not decrease the risk of a portfolio because of its high volatility. Therefore, an asset with a low correlation does not necessarily decrease the risk of an existing portfolio. This means it might be better to look at the covariances instead of the correlations, since these do take high volatility into account. However, since we will also be using the 2.5% CVaR as a risk measure, we will broadly define an asset a diversifier if it decreases the risk of the minimum risk portfolio. Nonetheless, we will also investigate the correlations and covariances of the assets for completeness.
One of the main reasons investors hold gold, is to protect their portfolio against the risk of high inflation. Research by Ghosh et al (2004), Levin et al. (2006), Conover et al. (2009) and Wang et al (2010) show that precious metal, and especially gold, serves as an effective hedge against inflation. Ghosh et al (2004) and Wang et al (2010) also show that the return on gold is highly correlated to CPI, and that the return on gold is equal to the inflation rate in the long run, meaning it will preserve purchasing power over a longer period of time. Another paper by Hammoudeh et al (2010) found that gold served as the best safe haven against inflation compared to other precious metals and stocks. Although, a paper by Johnson Soenen (1997) showed that the effectiveness of gold as a hedge against inflation is not constant over time.
Another reason for investors to hold gold, is because gold is uncorrelated or negatively correlated to most other assets, therefore making it a good diversifier. There has been ample research on the diversification benefits of gold. Early research by Sherman (1982), found a low correlation between equity and gold and found that adding 5% gold to a portfolio gave significant improvements. Later research by Jaffe (1989), suggested that adding gold to a portfolio can reduce portfolio risk, since the returns on gold have a low correlation with the returns of other assets. He investigated the pe-riod from September 1971 to June 1987, and found a correlation of 0.054 between the S&P 500 and gold. Besides that, he also calculated a beta of 0.09, which indicates that gold has little systematic risk. He therefore suggested that adding a 5% or 10% weight in gold to an equity portfolio both improves performance and reduces risk. Further research by Chau et al. (1990), tried to investigate whether the results of Jaffe (1989) were also valid for shorter time periods. They investigated two smaller time periods from Sept-1971 to Dec-1979, and from Jan-1980 to Dec-1988. Similar to Jaffe (1989), they found a correlation of 0.050 and a beta of 0.11 for the full time period. For the period from Sept-1971 to Dec-1979, they found a correlation of 0.011 and a beta of 0.03. For the period from Jan-1980 to Dec-1988, they found a correlation of 0.118 and a beta of 0.22. They therefore suggest that gold adds diversification benefits in the long and short-term and, adding 25% gold to equity portfolios.
period. Therefore suggesting that the optimal weight of gold in a portfolio differs per period. Smith (2002) also finds a low correlation between gold and European stocks in the short term, and finds no cointegration between gold and European stocks in the long term. Later research by Pulvermacher (2005), O’Connel (2005), Lucey et al (2006) and Ap Gwilym et al. (2010) confirm that gold has a low correlation to stocks and other asset classes. Moreover, they also found that adding gold will decrease the portfolios volatility. Additionally, Ap Gwilym et al. (2010) shows that the correlation between real interest rates and gold is negative. This implies that gold is positively correlated with bond prices, because bond prices are also negatively correlated with interest rates. Furthermore, research by McCownn Zimmerman (2006) and Michaud et al. (2006) confirm that gold has a beta which is approximately equal to 0.
Similarly, the safe haven properties of gold in extreme market conditions has been examined. Lawrence (2003), Pulvermacher (2005) and O’Connel (2005) show that in extreme market situa-tions, the correlation between gold and stocks are strongly negative. This makes gold an effective safe haven asset against extreme negative stock market returns. Further research by Baur and Lucey (2010) and Baur McDermott (2010), shows that in uncertain times gold is being bought by fearful investors, especially in periods of increased systematic risk like in the 2008 financial crisis. Because of these favorable characteristics of gold, many authors suggest allocating 5-25% of your portfolio into gold to reduce downside risk [Jaffe (1989), Chau et al. (1990) and Hillier et al. (2006)].
Bitcoin
Following the financial crisis of 2008, a paper published by Nakamoto (2008) introduced Bitcoin as a decentralized peer-to-peer payment system. He argued that fiat currencies are neither a good store of value, nor a good medium of exchange. He reasons that inflation erodes the purchasing power of money. Besides that, international payments in the current system can be very slow and expensive, due to all the financial mediators. Lastly, he argues that the current system excludes many individuals that have no access to a bank account. Contrary to fiat currency, Bitcoin has a fixed supply and is deflationary in nature. Also international payments are fast, cheap, and owning a Bitcoin wallet is accessible to everyone. These characteristics, he argues, would also make Bitcoin a great store of value.
Since the release of the paper by Nakamoto (2008), some papers have been published about the characteristics of Bitcoin. Early research by Glaser et al. (2014) concluded that the amount of Bitcoins being transferred using the Bitcoin network was significantly smaller compared to the volume being traded on exchanges. He showed that speculative demand was therefore the main driver for Bitcoin, not a medium of exchange. A paper by Dyhrberg (2016a) states that Bitcoin behaves similar to gold, since it returns no dividend or interest and its price is only driven by supply and demand. Hence why Bitcoin is sometimes being called the ’digital gold’ (Popper, 2015). Also, Corbet et al, (2018) argues that Bitcoin behaves like an asset, and might give diversification benefits because of its low correlation with other traditional assets.
et al (2017), investigated whether Bitcoin could act as a hedge or safe haven for the US dollar, bonds, oil, gold, commodity indexes, and the major stock market indices. They investigated the period from July 2011 to December 2015, and found that Bitcoin performs poorly as a hedge or safe haven, and is only useful for diversification because of its close to zero correlation between the main other asset classes.
Many later papers moreover suggest that Bitcoin could offer portfolio improvements because of its diversification benefits. For example, research by Guesmi et al (2018) from Jan-2012 to May-2018, showed that using Bitcoin in a portfolio of gold, oil and equities, did significantly improve the risk and reward of that portfolio due to its low correlation with other assets and high return. Another paper by Kajtazi Moro (2018) shows that Bitcoin improves the portfolio’s performance, however, they state that this effect is mainly through the increase in returns instead of a reduction of volatility. Later research by Platanakis Urquhart (2019), that use out of sample testing instead of backtesting, likewise finds that Bitcoin improves the risk-to-reward trade-off. However, again they find that this is mainly due to the high excess returns of Bitcoin. Later research by Symitsi et al (2019) finds that the correlation between Bitcoin and other assets is low, eventhough the overall risk of the portfolio does not decrease, because of the high volatility of Bitcoin. He does however find significant benefits of including Bitcoin in a portfolio. Again, because of the high returns Bitcoin can offer to a portfolio. This means that most current papers suggest that Bitcoin does not decrease the risk of the minimum risk portfolio, however they do find evidence that including Bitcoin gives a better risk-to-reward trade-off because of its high return.
2
Methodology
The techniques used in this thesis are based on the early work by Markowitz (1952) about portfolio optimization. Markowitz showed that owning an individual stock is more risky than owning multiple stocks. He then showed that while we can reduce the risk of a portfolio by diversifying over many stocks with the same return, this does not affect the return of this portfolio. Therefore, he proposed that optimizing a portfolio should be based on a risk-to-reward trade-off. The risk measure used by Markowitz was the variance of the returns, however recent research by Rockafellar & Uryasev (2002) propose using the conditional value at risk (CVaR). They argue that investors care more about the possible downside of their portfolio compared to the amount of variance their portfolio endures. To do this optimization we first fit a GARCH-EVT-COPULA model to historical data and simulate 5000 future return series based on this fitted model. To find the optimal portfolio we both use the traditional Mean-Variance optimization and the more recent Mean-CVaR optimization. This section starts of by explaining the relevant theory and literature on the GARCH-EVT-COPULA model, secondly it explains the different risk measures used to estimate the riskiness of a portfolio. Lastly it explains the step by step procedure used to estimate, simulate and evaluate the portfolio’s. The explained theory is based on McNeil et al. (2015) unless stated otherwise.
2.1 Univariate GARCH Models
are estimated using maximum likelihood and the optimal number of lags pp, qq are chosen by using AIC, BIC, SIC and HQIC. The most common model used in literature is the GARCH(1,1) model (Sahamkhadam et al. (2018)). However, we will also investigate GARCH models with lags up to (2,2) for completeness. The following definition of the GARCH model is used by McNeil et al. (2015).
Let pZtqtPZ be a SWNp0, 1q process. Where strict white noise (SWN) means that pZtq is i.i.d.
The process pXtqtPZ has mean µ with GARCHpp, qq errors if it satisfies the below conditions and
is covariance stationary. Xt“ µ ` σtZt (1) σ2t “ α0` p ÿ i“1 αipXt´i´ µq2` q ÿ j“1 βjσt´j2 (2)
where α0 ą 0, αi, βj ě 0, for i “ 1, . . . , p, j “ 1, . . . , q, and
řp
i“1αi`
řq
j“1βj ă 1.
The standardized residuals pZtqtPZobtained after fitting the GARCH model should be SWN(0,1).
This means the standardized residuals should not have any significant autocorrelation. Besides that there should not be any significant autocorrelation in the squared standardized residuals `Zt2˘tPZ, since this would indicate that the volatility has autocorrelation. To investigate these things formally a Ljung-Box test can be performed on both the raw and squared standardized residuals.
2.2 Behaviour of the Tail
A problem with modelling asset returns is that they exhibit fat tails. Our model can partly account for this by assuming that the standardized residuals pZtqtPZ are t-distributed or skewed
t-distributed instead of normally distributed. However, Mc Neil et al. (2015) and Sahamkhadam et al. (2018) argue that the best fit is found by assuming that the standardized residuals are modelled using a gaussian kernel in the center of the distribution and modelled using a generalized Pareto (GPD) distribution in the upper and lower tails of the distribution. The following model is used by Sahamkhadam et al. (2018): Fipxiq “ $ ’ ’ ’ & ’ ’ ’ % NuL N ´ 1 ` ξL uL´xj βL ¯´ 1 ξL , xj ă ´uL φ pxjq , uLă xj ă uR 1 ´NuR N ´ 1 ` ξL un´xj βR ¯´ 1 ξR , xj ą uR (3)
where ξ, β, µL, µR, NuLand NuRare the shape, scale, lower threshold, upper threshold, number of observations below lower threshold and number of observations above upper threshold respectively, and φp¨q is the Gaussian distribution function. After obtaining the standardized residuals pZtqtPZ
2.3 Dependence and Copula Models
Traditionally the dependence of variables was estimated by calculating a scalar dependence mea-sure for the pair of random variables pX1, X2q. One of the commonly used dependence measures is
the Pearson linear correlation coefficient denoted by: ρX1,X2“ CovpXσ 1,X2q
X1σX2 . The problem with this
approach is that it only gives us an estimate for the average linear dependence between the vari-ables. So for example, it does not give us information whether the dependence between variables is larger in the tails of the distribution compared to the center. However, we know from empirical observations that asset returns will become more dependent when large negative returns occur. To get more information about the dependence structure we can also look at the rank correla-tion and tail dependence of our assets. To calculate the rank correlacorrela-tion we need to consider an independent copy p ˆX1, ˆX2q of our original pair pX1, X2q. We can then define the Kendall’s tau
correlation coefficient by: EpsignppX1´ ˆX1qpX2´ ˆX2qqq. A high positive rank correlation therefore
means that when X1ą ˆX1 we will also see X2ą ˆX2 more often.
Lastly we can look at the upper and lower tail dependence coefficient. The upper tail dependence coefficient is defined in the following way: P rpX1 ą qα|X2ą qαq where qαis the α-th quantile of the
data. The lower tail dependence coefficient is defined similarly as follows: P rpX1 ă qα|X2 ă qαq.
The appropriate quantiles are chosen in such a way that we are sufficiently far into the tail while still having sufficient data points to estimate the coefficient. In this thesis we choose to use the 10% level for the tails, since choosing a level further into the tail will result in a lack of observations. The previous dependence measures give only one single scalar value for the dependence between two variables. However, the dependence between two assets is most likely non linear. To solve this problem we can use a copula modelling approach which lets us separate the marginal behaviour from the dependence structure. This allows us to estimate the dependence between variables over its entire domain. So for example, we can have that variables are highly correlated in the tail and show no correlation in the center of the distribution. We denote the distribution function Cpuq “ C pu1, . . . , udq as a d-dimensional copula on r0, 1sd with standard uniform marginals. The
following properties must hold for a multivariate distribution to be a copula (McNeil et al. 2015). 1. C pu1, . . . , udq is increasing in each component ui
2. C p1, . . . , 1, ui, 1, . . . , 1q “ ui for uiP r0, 1s and for each i P t1, . . . , du
Using the definition of a copula (Sklar 1959) proved that for a multivariate distribution function F with margins F1, ..., Fdthere exists a d-dimensional copula such that:
F px1, . . . , xdq “ C pF1px1q , . . . , Fdpxdqq (4)
This means that every multivariate distribution F px1, ..., xdq can be written as a copula Cpu1, ..., udq
with marginals Fipxiq instead of ui for i “ 1, 2, .., d. If we evaluate this equation at the arguments
xi “ Fi´1puiq, i “ 1, ..., d we get the following equation:
F`F1´1pu1q , . . . , Fd´1pudq
˘
“ C pu1, . . . , udq (5)
The first equation tell us that we can always construct a multivariate distribution F from the marginals F1, ..., Fd and a copula C pu1, . . . , udq. This means we can individually fit marginal
shows that we can obtain the copula from a multivariate distribution with continuous margins. This means that the dependence structure of the multivariate distribution is captured by the copula. Therefore, the copula perfectly captures the non-linear dependence between the variables. We can now find the density of the copula by differentiating equation (5) in the following way:
c pu1, . . . , udq “ BdC pu1, . . . , udq Bu1, . . . , Bud “ f`F ´1 1 pu1q , . . . , Fd´1pudq ˘ śd i“1fi`F ´1 i puiq ˘ (6)
Similarly, we can also find the density of f by differentiating equation (4):
f px1, . . . , xdq “ BdC pF1px1q , . . . , Fdpxdqq Bx1, . . . , Bxd “ c pF1px1q , . . . , Fdpxdqq d ź i“1 fipxiq (7)
Using the results of (6) and (7) it is possible to derive the log likelihood function used to estimate the copula parameters. The log likelihood function is given by:
l px1, x2, . . . , xdq “ řn i“1log pc pF1pxi1q , F2pxi2q , . . . , Fdpxidq |θqq ` řn i“1 řd j“1log pfjpxijqq (8) The log likelihood function consists of two parts. The first part optimizes the copula function. The second part optimizes the marginal distributions. Therefore the optimization optimizes the marginals and the dependence structure separately to obtain the multivariate distribution.
Many copulas can be used to estimate the dependence structure of a multivariate distribution. This thesis makes use of the Gauss and student-t copulas following the results of the paper by Sahamkhadam et al. (2018). He found that the Gauss and student-t copulas performed best in modelling the the dependence structure of stocks. Another paper by Righi et al. (2015) showed that the student-t copulas are best at modelling the dependence structure of bonds. The main advantage of the Gauss and t-copula is that it is possible to estimate different correlations between the marginals. The key advantage of the t-copula is that it allows for fat tails of the dependence structure. Therefore assets can be modelled to have higher dependence in the tails compared to the center of the distribution. This property of the students-t copula is empirically very well suited for financial assets. We can define the Gauss copula in the following way:
CPGapuq “ ΦP `Φ´1pu1q , . . . , Φ´1pudq
˘
(9) Where the univariate normal distribution is denoted by Φ and the multivariate normal distribution is denoted by ΦP, with P being the corresponding correlation matrix of the copula. Similarly we
can define the the t copula by:
Cv,Pt pu1, . . . , udq “ tv,P`t´1v pu1q , . . . , t´1v pudq
˘
(10) where tv is the univariate t distribution and tv,P the multivariate t distribution with correlation
2.4 Risk Measures
Like previously stated we want to optimize our portfolio using a risk-to-reward trade-off. We mainly focus on the Mean-CVaR optimization, however we still want to include the Mean-Variance approach for completeness. In the models given below we have that ri is the return for asset i and
rp “řNi“1wiri is the return of the portfolio with wi being the weight of asset i. In section 2.5 we
will define more detailed notation and also use this notation in the given models below. For now, the Mean-Variance model is given by:
minimize w1Σw
wiě 0 for all i “ 1, 2, .., N
řN
i“1wi“ 1
rp ě A
Where w is the weight vector N x1 and Σ is the N xN covariance matrix of the returns. The fourth restriction is optional and specifies that the minimum return of our portfolio should be A. So we simply want to minimize the variance with respect to the weights given that the return on the portfolio exceeds A.
Recently investors are more interested in metrics like the value at Risk (VaR). Therefore, this metric is increasingly being used in portfolio optimization (Consigli, 2002). The value at risk gives a value which is not exceeded with probability 1 ´ α. So for example, when we have a VaR0,01 of
´12% we know that 99% of the returns are above ´12%. This risk measure is now commonly used by financial regulators and investors to measure the downside risk of portfolios. However, Rockafel-lar & Uryasev (2002) argued that the Value at risk does not satisfy the sub-additivity condition. A risk measure ρ satisfies the sub-additivity condition when ρpXq ` ρpY q ě ρpX ` Y q for two port-folios X and Y. Therefore, a merger of two portport-folios cannot increase the total risk. Besides that, the value at risk only gives a measure for the α-th quantile of the returns, however it says nothing about the distribution of these extreme returns. Therefore, they argued that the conditional value at risk (CVaR) is a superior risk measure. The CVaR measures the expected return given that the α ´ th value at risk is exceeded. This risk measure does satisfy the sub-additivity condition and will therefore be used in this thesis. The optimization problem using the CVaR becomes:
maximize Errp|rpď qαs
wiě 0 for all i “ 1, 2, .., N
řN
i“1wi“ 1
rp ě A
Where qα is the α-th quantile of the portfolio returns rp. So we take the expectation of the
portfolio returns given they are below the threshold qα. This gives us the expected loss in an
extreme negative situation. In this thesis we will be mainly using the 2.5% CVaR, however we will also investigate whether our results are robust when we change to a 1% or 5% CVaR level.
will be using our own expected return assumptions. For bonds and stocks we take these return assumptions from a report by Robeco (2019) about their future expected return assumptions for the coming five years. For gold we will be using the European inflation rate given by Robeco (2019). We do this because Ghosh et al (2004) showed that the return on gold is equal to the inflation rate in the long run. Lastly, we will be trying out a range of different expected return assumptions for Bitcoin.
Using these expected return assumptions we can transform our simulated return series into re-turn series that still have the same simulated volatility and correlation structure, but with our new expected returns. For example, if our scenarios give a expected yearly return of 5% for stocks, but the Robeco return assumptions give it only a 4% return, we can solve this by subtracting 1/52% from all stock returns. Doing this our simulations will also have an expected return of 4%, without changing the volatility and correlation structure of our simulations. We repeat this procedure for all our asset classes.
When we have adjusted the simulated returns to have the same expected return as our expected return assumptions we can obtain the statistics required for the optimization. We can calculate the expected value, standard deviation and CVaR of the cumulative portfolio returns for different asset weights. To find the optimal weights we will be using Generalized Reduced Gradient decent, which looks at the gradient of our objective function with respect to the weights. We can then adjust the weights according to the gradient and calculate the gradient in the new situation. This process will repeat itself iteratively until the improvements to the objective function are below a treshold . To make the solution robust we will be using a random distributed population of initial values to obtain the global minimum or maximum instead of a local minimum or maximum.
2.5 Procedure
This section will give an overview of the procedure used to fit the correct model, simulate the new returns and analyze these simulated returns. The Procedure used is similar to the procedure used by Sahamkhadam et al. (2018). The following part will give an overview of this procedure divided into eight steps. The total amount of assets we investigate in this thesis is five and we will be using log-returns.
1. Estimate the amount of lags pp, qq used in the GARCH model. We do this by using the AIC, BIC, SIC, HQIC information criteria. We do this for all five assets.
2. Fit a GARCHpp, qq model with skewed-t distribution innovations to the univariate return series using maximum likelihood. We obtain the standardized residuals ˆZt “ p ˆZ1,t, ˆZ2,t, ..., ˆZ5,tq
which should be SWNp0, 1q. We can check whether they are i.i.d. by formally testing the stan-dardized residuals and squared stanstan-dardized residuals using the Ljung-Box test.
3. Fit the semi-parametric function given by equation p3q to the standardized residuals obtained in step 2. The upper and lower 10% tail are modelled like a Generalized Pareto Distribution and the interior is modelled using a Gaussian kernel. We can now use this distribution to perform a probability integral transformation (PIT) in the following way:
4. Use Sklar’s theorem and insert the uniform variables ˆutj into equation 5. Using maximum
likelihood we can now obtain the parameter vector ˆθ. We can now also derive the multivariate distribution by using the marginals and the copula.
p F pˆut1, ˆut2, . . . , ˆut5q “ pC ´ p F1pˆut1q , pF2pˆut2q , . . . , pFdpˆut5q |pθ ¯ (12) 5. Simulate 5000 uniformly distributed random series wi,t,j for i “ 1, 2, ..., 5000, t “ 1, ..., 260
and j “ 1, .., 5. Insert these random series in the estimated copula to get 5000 uniformly distributed series with the estimated dependency structure.
p
vi,t “ ppvi,t,1,pvi,t,2, . . . ,pvi,t,5q
“ pC ´
p
F1pwi,t,1q , pF2pwi,t,2q , . . . , pF5pwi,t,5q |pθ
¯
,vpi,t,j „ U p0, 1q
(13) 6. Use the inverse of the semi-parametric distribution estimated at step 3 to calculate the new standardized residuals:
y
xi,t“ pˆxi,t,1, ˆxi,t,2, . . . , ˆxi,t,5q
“ ´
ˆ F´1
1 ppvi,t,1q , ˆF2´1ppvi,t,2q , . . . , pFd´1ppvi,t,5q
¯
, i “ 1, . . . , 5000, t “ 1, . . . , 260 (14) 7. Fill in these standardized residuals into our fitted GARCH(p,q) model to get the simulated returns ri,t,j for t “ 1, 2, ..., 260, i “ 1, 2, .., 5000 and j “ 1, 2, .., 5. We now have 5000 scenarios
of 260 weekly returns for each of our five assets. Because we are using log-returns we can sum these returns over time to get the five year cumulative returns Cri,j “
ř260
t“1ri,t,j for each scenario
and each asset. We can now calculate the expected cumulative return ACrj “
ř5000
i“1 Cri,j
5000 of all
the simulations for j “ 1, 2.., 5. Let the five year log expected return assumptions obtained from Robeco (2019) be denoted by ACrjRfor j “ 1, 2, .., 5. We can now adjust our simulated returns by: adj ´ ri,t,j “ ri,t,j ´
ACrj´ACrRj
260 . From this point forward we will just use the original notation for
this adjusted series. So we again have Cri,j “
ř260
t“1ri,t,j for the cumulative adjusted returns for
each asset. Lastly, the portfolio returns are obtained by: Crpi “
ř5
j“1wdCri,j for i “ 1, 2, .., 5000.
So we end up with 5000 portfolio cumulative returns for each choice of weights wj for j “ 1, 2, .., 5.
The expected value of these returns is given by ErCrps “ ACrp “
ř5000
i“1 Crpi
5000 .
8. We now have all the tools to optimize our objective functions given in section 2.4. For the Mean-Variance model we want to optimize the following objective function:
minimize
w1,..,w5
5000
ÿ
i“1
pCrpi´ ErCrpsq2, such that ErCrps ě A.
So we want to minimize the variance of the five year cumulative portfolio returns with respect to the weights given that the expected return of the portfolio exceeds a certain threshold A. Besides the Mean-Variance approach we can optimize using CVaR in the following way:
maximize
w1,..,w5
ErCrp|Crp ă qαs, such that ErCrps ě A.
choose many starting points pw1, w2, ..., w5q and look at the gradient of the objective function with
respect to the weights. We can then adjust the weights according to the gradient and calculate the gradient in the new situation. This process will repeat itself iteratively until the improvements made to the objective function are below a threshold .
3
Data Analysis
The data used consists of the weekly log returns of Bitcoin, gold, stocks, government bonds and corporate bonds from the period 7 April 2013 until 22 March 2020. The date of 14 April 2013 was the earliest moment that the price of Bitcoin exceeded $100. This gave Bitcoin a lot of media attention and legitimacy. Therefore, we choose to use this starting date for our research. We use weekly data since bitcoin is traded seven days a week while traditional assets are only traded five days a week. The use of weekly data is in line with many other papers about Bitcoin in the past (Platanakis Urquhart, 2019). For the price of Bitcoin and gold we use the weekly spot price of both these assets. Stocks are tracked by using the Vanguard Total World Stock Index (VT). This index tracks the world stock market and is similar to the equity portfolio exposure of a typical European investor. The allocation of this index is approximately 85% in developed markets and 15% in developing markets. The government bond index used is the iShares Core Government Bond UCITS (EUNH) which tracks euro denominated bonds. The corporate bond index used is the iShares Core Corporate Bond UCITS (EUN5), which also tracks euro denominated bonds. We have converted all prices in euros using the EUR/USD exchange rate at the time and we do not hedge any exchange rate risk. All the historical data is obtained from the investing.com website. We now give some additional information about the bond indexes used. The government bond index (EUNH) has a duration of 8.38 and currently has the following geographical distribution: France 24.30%, Italy 22.04%, Germany 18.17%, Spain 14.20%, Belgium 6.09%, Netherlands 4.67%, Austria 3.62%, Portugal 2.18%, Ireland 1.78%, Finland 1.41% and other 1.55%. Therefore, this index does have a high allocation to somewhat riskier countries like Italy, Spain and France. The credit quality of the bonds have the following distribution: AAA Rated 22.82%, AA Rated 35.40%, A Rated 17.11% and BBB Rated 24.54%. Besides that, the corporate bonds index (EUN5) has a duration of 5.20 and currently has the following credit quality distribution: AAA Rated 0.41%, AA Rated 10.01%, A Rated 36.49% and BBB Rated 53.02%. Therefore, both the government and corporate bond indexes consist of a combination of low risk bonds and higher risk investment grade bonds.
3.1 Preliminary Analysis
exchange rate has greatly decreased in a short period during the European debt crisis, however it has remained relatively stable during the corona crisis.
Bitcoin has seen an explosive bullish rally from 2013 to the start of 2018. The market price of one bitcoin went from 100$ to nearly 20,000$. However, Bitcoin was unable to hold onto these gains and has seen its price crashing to 3000$. Since then it has been consolidating between these levels. Before the corona crisis it again reached a price of 10,500$, however the price crashed to 3800 during the corona crisis. It currently holds a price of around 5500$. This latest crash suggests that Bitcoin is not the safe heaven asset people give it credit for. Instead it is a high risk speculative asset that gets sold of when a crisis occurs. Two things to note about the price movement in Bitcoin is that the market suffers from a lack of liquidity. Secondly, Bitcoin historically has had rally’s of approximately 10,000% followed by a drop of 85%.
40 50 60 70 2014 2016 2018 2020 date Pr ice (a) Stocks 110 115 120 125 130 135 2014 2016 2018 2020 date Pr ice (b) Government bonds 120 125 130 135 2014 2016 2018 2020 date Pr ice (c) Corporate bonds 1000 1200 1400 2014 2016 2018 2020 date Pr ice (d) Gold 0 5000 10000 15000 2014 2016 2018 2020 date Pr ice (e) Bitcoin 1.1 1.2 1.3 1.4 2014 2016 2018 2020 date Eur/USD (f) EUR/USD
Figure 1: Market Prices in Euro
the risk-to-reward trade-off for investors. Lastly, we find that gold is relatively unaffected by the corona crisis. The table also shows the skewness, kurtosis and the results of the Jarque-Bera test. We can see that the skewness of our assets is more negative when we include the corona crisis, and the kurtosis is generally higher. This is because the corona crash introduces a few very high negative returns into the data set. Lastly, we can also see the effect of these changes on the value of the Jarque-Bera statistic. The value is generally higher, however we also reject the null hypothesis of normality when we exclude the corona crisis.
Table 1: Descriptive statistics of weekly returns including corona crisis. Mean Std.dev Skewness Kurtosis Jarque Bera
Bitcoin 1.230% 13.75% 0.212 8.557 469.71˚˚˚
Gold 0.074% 1.833% -0.350 4.512 42.00˚˚˚
Stocks 0.060% 2.261% -1.406 8.035 503.03˚˚˚
Gov-Bonds 0.045% 0.564% -1.121 7.785 422.29˚˚˚
Corp-Bonds 0.002% 0.480% -3.383 29.58 11378˚˚˚
*** means p value of ă 0.01; returns are in log returns.
Table 2: Descriptive statistics of weekly returns excluding corona crisis. Mean Std.dev Skewness Kurtosis Jarque Bera
Bitcoin 1.362% 13.52% 0.368 8.607 474.34˚˚˚
Gold 0.068% 1.743% -0.213 4.128 21.56˚˚˚
Stocks 0.174% 2.006% -0.679 4.731 71.82˚˚˚
Gov-Bonds 0.053% 0.535% -0.593 4.622 59.87˚˚˚
Corp-Bonds 0.025% 0.374% -0.443 6.133 157.29˚˚˚
*** means p value of ă 0.01; returns are in log returns.
Table 3: Pearson correlation including corona crisis. Bitcoin Gold Stocks Gov-Bonds Corp-Bonds
Bitcoin 1.000 0.027 0.077 0.120 0.146
Gold 1.000 0.061 0.333 0.266
Stock 1.000 0.098 0.399
Gov-Bonds 1.000 0.617
Corp-Bonds 1.000
Table 4: Pearson correlation excluding corona crisis.
Bitcoin Gold Stocks Gov-Bonds Corp-Bonds
Bitcoin 1.000 -0.034 0.038 0.060 0.120
Gold 1.000 0.001 0.295 0.237
Stocks 1.000 0.023 0.213
Gov-Bonds 1.000 0.675
Corp-Bonds 1.000
Tables 5 and 6 show the corresponding Kendal’s tau rank correlations. A first thing to note is that the Kendal’s tau correlations are lower compared to the Pearson correlations. This suggests that the correlation between the assets is not linear. There is more correlation in the tails, which increases the Pearson correlation more proportionally compared to the rank correlation. This is because the rank correlation only looks at the signs and Pearson correlation uses the high covariance in the tail disproportionally much to estimate the correlation coefficient. For example, we can see that the Pearson correlation of stocks and corporate bonds is significantly higher when we include the corona crisis. However the Kendal’s tau rank correlations are similar for both periods.
Table 5: Kendal’s tau rank correlation including corona crisis. Bitcoin Gold Stocks Gov-Bonds Corp-Bonds
Bitcoin 1.000 0.032 0.027 0.030 0.053
Gold 1.000 0.007 0.172 0.167
Stocks 1.000 0.018 0.178
Gov-Bonds 1.000 0.479
Corp-Bonds 1.000
Table 6: Kendal’s tau rank correlation excluding corona crisis. Bitcoin Gold Stocks Gov-Bonds Corp-Bonds
Bitcoin 1.000 0.018 0.024 0.028 0.048
Gold 1.000 0.009 0.166 0.160
Stocks 1.000 0.014 0.170
Gov-Bonds 1.000 0.491
Corp-Bonds 1.000
the large variance of Bitcoin compared to the other assets gives a high covariance regardless. This shows that an asset does not necessarily reduce the risk of a portfolio when it has a low correlation.
Table 7: Covariance including the corona crisis.
Bitcoin Gold Stocks Gov-Bonds Corp-Bonds
Bitcoin 1.89e-02 6.87e-05 2.38e-04 9.33e-05 9.60e-05
Gold 3.36e-04 2.54e-05 3.44e-05 2.34e-05
Stocks 5.11e-04 1.25e-05 4.33e-05
Gov-Bonds 3.18e-05 1.67e-05
Corp-Bonds 2.30e-05
Table 8: Covariance excluding the corona crisis.
Bitcoin Gold Stocks Gov-Bonds Corp-Bonds
Bitcoin 1.83e-02 -7.88e-05 1.04e-04 4.32e-05 6.05e-05
Gold 3.04e-04 4.11e-07 2.75e-05 1.54e-05
Stock 4.02e-04 2.50e-06 1.60e-05
Gov-Bonds 2.86e-05 1.35e-05
Corp-Bonds 1.40e-05
Tables 9,10,11 and 12 show the upper and lower tail dependence between the variables for both the periods including and excluding the corona crisis. The threshold for the lower tail is the 0.10 quantile, and similarly the threshold for the upper tail is the 0.90 quantile. The upper tail depen-dence measures the probability that X2 has an observation in the upper tail given that X1 has an
observation in the upper tail. The value we expect to see when there exists no tail dependence is approximately 0.10 or 10%. Values that are significantly larger than 0.10 have tail dependence. We took the 10% quantiles because of a lack of data for quantiles further in the tail. This is because we only use the 10% most extreme weekly returns to test the tail dependence. This gives us 36 observations in the tail when we include the corona crisis and 35 observations in the tail when we exclude the corona crisis. This is the absolute minimum amount of observations to give any reliable evidence for tail dependence. When there exists no tail dependence we expect to find 3.6 of the 36 observations to be in the tail. We use the binomial distribution to test significance and we find a p value smaller than 0.10 when we find more than 7 observations in the tail.
are unexpected by investors. When we look at the case excluding the corona crisis we can see that the tail dependences are very similar. However, when we exclude the corona crisis most pairs have a slightly lower tail dependence compared to the case when we include the corona crisis.
Table 9: Upper tail dependence including corona crisis (N=36). Bitcoin Gold Stocks Gov-Bonds Corp-Bonds
Bitcoin 1.000 0.138 0.193˚ 0.055 0.138 Gold 1.000 0.110 0.193˚ 0.193˚ Stocks 1.000 0.165 0.220˚˚ Gov-Bonds 1.000 0.386˚˚˚ Corp-Bonds 1.000 P rpX1 ą q0.90|X2ą q0.90q, ˚ ˚ ˚ “ p ă 0.01, ˚˚ “ p ă 0.05, ˚ “ p ă 0.10
Table 10: Lower tail dependence including corona crisis (N=36). Bitcoin Gold Stocks Gov-Bonds Corp-Bonds
Bitcoin 1.000 0.138 0.193˚ 0.110 0.248˚˚˚ Gold 1.000 0.138 0.358˚˚˚ 0.248˚˚˚ Stocks 1.000 0.138 0.193˚ Gov-Bonds 1.000 0.496˚˚˚ Corp-Bonds 1.000 P rpX1 ă q0.10|X2ă q0.10q, ˚ ˚ ˚ “ p ă 0.01, ˚˚ “ p ă 0.05, ˚ “ p ă 0.10
Table 11: Upper tail dependence excluding corona crisis (N=35). Bitcoin Gold Stocks Gov-Bonds Corp-Bonds
Bitcoin 1.000 0.140 0.112 0.056 0.140 Gold 1.000 0.112 0.196˚ 0.169 Stocks 1.000 0.169 0.225˚˚ Gov-Bonds 1.000 0.393˚˚˚ Corp-Bonds 1.000 P rpX1 ą q0.90|X2ą q0.90q, ˚ ˚ ˚ “ p ă 0.01, ˚˚ “ p ă 0.05, ˚ “ p ă 0.10
Table 12: Lower tail dependence excluding corona crisis (N=35). Bitcoin Gold Stocks Gov-Bonds Corp-Bonds
4
Results
This section starts of with giving the results of the fitted GARCH-EVT-COPULA model. The results of the log likelihood and information criterion’s for a number of GARCHpp, qq models are listed in table 13. The results are based on the model including the corona crisis. The results show that adding more than one lag increases the log likelihood slightly. However, when taking into account the information criterion’s penalty attributed to adding lags, we can see that the GARCH(1,1) model fits best. This result is similar to the findings of Sahamkhadam et al. (2018). Therefore, we choose to fit a GARCH(1,1) model to our univariate series for both the cases including and excluding the corona crisis.
Table 13: Information criterion results including the corona crisis.
GARCH order Loglikelihood AIC BIC SIC HQIC
0,1 5190.87 -28.399 -28.013 -28.416 -28.246 1.0 5228.78 -28.610 -28.224 -28.628 -28.457 1,1 5267.40 -28.796 -28.356 -28.818 -28.621 1,2 5266.91 -28.765 -28.272 -28.793 -28.569 2,1 5267.80 -28.770 -28.277 -28.798 -28.574 2,2 5270.35 -28.757 -28.210 -28.790 -28.539
The parameter estimates of the GARCH(1,1) models with and without the corona crisis are summarized in table 14 and 15. We can see that all the models have a very low α0, which implies
that no significant constant term is added to the volatility. Furthermore, we see that all the models have a high and significant β1. This means that volatility is very persistent, because β1 is the
coefficient of the lagged volatility term. For Bitcoin and stocks we also see a high and significant α1. This implies that large shocks in the previous period will increase the volatility in this period.
Table 14: Univariate GARCH results including the corona crisis.
µ α0 α1 β1 Skew Shape LB Zt LB Zt2
Bitcoin 6.08e-03 1.68e-03 0.337˚˚ 0.638˚˚˚ 0.948˚˚˚ 3.849˚˚˚ 1.489 0.041
Gold 7.92e-04 6.52e-06 0.082 0.903˚˚˚ 0.896˚˚˚ 8.557˚˚ 0.826 3.004
Stocks 1.52e-03 2.12e-05 0.174˚˚˚ 0.802˚˚˚ 0.761˚˚˚ 6.042˚˚˚ 0.284 0.594
Gov-Bonds 5.66e-04 1.26e-06 0.043 0.922˚˚˚ 0.846˚˚˚ 4.557˚˚˚ 0.024 0.052
Corp-Bonds 5.10e-05 6.61e-07 0.069 0.910˚˚˚ 0.672˚˚˚ 3.448˚˚˚ 2.908 2.060
˚˚ “ p ă 0.05, ˚ ˚ ˚ “ p ă 0.01
Table 15: Univariate GARCH results excluding the corona crisis.
µ α0 α1 β1 Skew Shape LB Zt LB Zt2
Bitcoin 7.87e-03 1.52e-03 0.314˚˚ 0.636˚˚˚ 0.976˚˚˚ 4.423˚˚˚ 2.328 0.114
Gold 1.05e-03 3.01e-06 0.047 0.940˚˚˚ 0.921˚˚˚ 10.31˚˚ 3.730 2.275
Stocks 1.70e-03 1.96e-05 0.138˚˚˚ 0.822˚˚˚ 0.773˚˚˚ 7.223˚˚˚ 1.801 0.082
Gov-Bonds 6.05e-04 1.55e-06 0.050 0.899˚˚˚ 0.859˚˚˚ 5.517˚˚˚ 0.474 0.287
Corp-Bonds 1.74e-04 4.70e-07 0.049 0.919˚˚˚ 0.700˚˚˚ 4.385˚˚˚ 0.979 0.019
˚˚ “ p ă 0.05, ˚ ˚ ˚ “ p ă 0.01
The results of the Generalized Pareto distribution fits for the tails including and excluding the corona crisis are given in table 16 and 17. The tails are fitted using maximum likelihood and the first three columns show the fit in the upper tails. With ξ being the shape parameter and β the scale parameter. Besides that, the cut off point of the tails is given by u. The following three columns show the results for the lower tails. Interpreting these parameters is hard since both the shape, scale and cut off parameters affect each other. The size of the scale parameter fixes where the distribution starts, so a large value means the entire distribution is shifted to the right. The shape parameter fixes how fast the probability density function converges to zero, and the threshold parameter shows the least extreme value used to fit the GPD distribution. Therefore, either having a high ξ or β or u means that more of the probability mass is located further in the tails. Looking at the results we can see that Bitcoin has a significantly higher scale parameter in the upper tail compared to the other assets. However, it also got a lower shape parameter, so the distribution is shifted more to the right and from that point onward it converges to zero faster. Nonetheless, since the difference in the scale parameter is significantly higher compared to the difference in the shape parameter, we find that Bitcoin has more probability mass far in the upper tail compared to the other assets. In the lower tail the story is a little different. There we find that stocks and corporate bonds have more probability mass far in the lower tail. However, much of this difference fades when we look at the results excluding the corona crisis.
Table 16: Generalized Pareto Distribution parameters including corona crisis.
Table 17: Generalized Pareto Distribution parameters excluding corona crisis. Asset ξU βU uU ξL βL uL Bitcoin -0.189 0.792 1.172 -0.253 0.949 -1.048 Gold 0.119 0.401 1.296 -0.273 0.848 -1.273 Stocks -0.033 0.403 1.146 -0.181 0.933 -1.332 Gov-Bonds 0.020 0.497 1.125 -0.425 1.263 -1.135 Corp-Bonds 0.256 0.354 1.015 -0.073 0.947 -1.167
From these models we have simulated two data sets. Each consisting of 5000 scenarios for a period of 260 weeks. Some descriptive statistics of these simulated data sets are displayed in the first three columns of tables 18 and 19. First the annualized geometric mean return for the five year period is given. The next column shows the annualized geometric standard deviation. We can see that the return of Bitcoin is very high compared to the other assets. However, Bitcoin also has a very large annualized standard deviation. Especially, when we include the corona crisis in our data set. One reason for the higher volatility when including the corona crisis is the higher volatility in the historical data. However, another reason is that our GARCH model has volatility memory. Therefore, the current volatility is important for the volatility of the simulated returns. Including the corona crisis will therefore give us higher volatility simulations, especially in the first year. We also see this property in the annualized standard deviations of the other asset classes. They are all lower when we exclude the corona crisis from our dataset. Besides the lower standard deviation we can also see that including the corona crisis in our dataset decreases the mean return of all our assets except for gold. The next column shows the 2.5% conditional value at risk (CVaR) for the five year cumulative returns. So this shows the five year cumulative expected loss for the 2.5% worst scenarios. We can see that including the corona crisis makes the expected loss higher. However, for both time periods we can see that Bitcoin has the highest risk. Followed by stocks, then gold and bonds. Lastly, we can see that government bonds are less risky compared to corporate bonds when we use the CVaR as our risk measure instead of variance.
by taking 85% of the future expected returns of developed markets and 15% of developing markets. This allocation is equal to the allocation of the stock index used. Furthermore, we assume that the return on gold is equal to the European inflation rate (Ghosh et al. (2004). The European inflation rate is expected to be 1.6% for the coming 5 years (Robeco 2019). However, throughout our entire analysis, we will try out different returns on gold, since this is one of the two main assets we want to analyze. For the return on Bitcoin we will also be using multiple return assumptions throughout this analysis. Lastly, we have shown the new standard deviation and new 2.5% Cvar in columns five and six. Note that the standard deviation does not change when we change the return assumptions.
Table 18: Results simulated returns including the corona crisis.
Mean Std.dev 5 year CVaR New Mean New Std.dev New CVaR
Bitcoin 73.10% 338.4% -99.39% variable 338.4% variable
Gold 5.283% 16.03% -48.48% 1,600% 16.02% -56.89%
Stocks 2.941% 33.25% -86.74% 3.325% 33.24% -86.49%
Gov-Bonds 2.707% 4.152% -10.55% -0.375% 4.152% -23.19%
Corp-Bonds 0.237% 4.254% -26.72% 0.250% 4.254% -26.68%
Geometric returns are used. Mean and Std.dev are annualized. 2.5% CVaR of the 5 year cumulative returns is displayed.
Table 19: Results simulated returns excluding the corona crisis.
Mean Std.dev 5 year CVaR New Mean New Std.dev New CVaR
Bitcoin 91.76% 182.1% -91.41% variable 182.1% variable
Gold 5.233% 10.53% -26.44% 1,600% 10.53% -38.29%
Stocks 7.186% 17.52% -50.98% 3.325% 17.52% -59.19%
Gov-Bonds 3.017% 3.871% -5.932% -0.375% 3.871% -20.431%
Corp-Bonds 1.124% 2.605% -8.519% 0.250% 2.605% -12.41%
Geometric returns are used. Mean and Std.dev are annualized. 2.5% CVaR of the 5 year cumulative returns is displayed.
restriction. Therefore, we need to be careful with making strong conclusions about the results, because the results rely heavily on the return assumptions we use. Excluding the corona crisis does not have a great impact on the weight of stocks and bonds in the portfolio. However, it replaces the governments bonds in the optimal portfolio by corporate bonds. This is mainly due to the high volatility in corporate bonds during the corona crisis.
Table 20: Benchmark portfolio minimizing Std.dev. including corona crisis. Minimum yearly return Std.dev Stocks Gov-Bonds Corp-Bonds
None 3.767% 0.000% 53.08% 46.92%
0% 3.791% 0.689% 43.24% 56.05%
1% 8.590% 24.67% 0.000% 75.33%
2% 18.35% 57.28% 0.000% 42.72%
3% 29.42% 89.57% 0.000% 10.43%
The last three columns show the portfolio weights and sum up to 100%.
Table 21: Benchmark portfolio minimizing Std.dev. excluding corona crisis. Minimum yearly return Std.dev Stocks Gov-Bonds Corp-Bonds
None 2.605% 0.000% 0.000% 100.0%
0% 2.605% 0.000% 0.000% 100.0%
1% 4.941% 24.67% 0.000% 75.33%
2% 10.04% 57.28% 0.000% 42.72%
3% 15.63% 89.57% 0.000% 10.43%
The last three columns show the portfolio weights and sum up to 100%.
In practice investors are more interested in minimizing their downside risk compared to min-imizing the volatility of the returns. Therefore, we mainly focus on optmin-imizing the CVaR in this thesis. Tables 22 and 23 show the results of this optimization. We can see that the weights are very similar, especially for the 1,2 and 3% minimum return restrictions we get exactly the same allocations. The main question we want to answer in the coming sections is whether we can con-struct portfolios consisting of gold, Bitcoin, stocks and bonds that are less risky compared to the benchmark portfolios given here.
Table 22: Benchmark portfolio maximizing Cvar including corona crisis. Minimum yearly return 2.5% CVaR Stocks Gov-Bonds Corp-Bonds
None -21.72% 0.748% 67.92% 31.34%
0% -22.19% 2.460% 51.81% 45.73%
1% -42.94% 24.67% 0.000% 75.33%
2% -68.78% 57.28% 0.000% 42.72%
3% -83.41% 89.57% 0.000% 10.43%
Table 23: Benchmark portfolio maximizing Cvar excluding corona crisis. Minimum yearly return 2.5% CVaR Stocks Gov-Bonds Corp-Bonds
None -12.37% 0.594% 0.000% 99.41%
0% -12.37% 0.594% 0.000% 99.41%
1% -22.05% 24.67% 0.000% 75.33%
2% -40.65% 57.28% 0.000% 42.72%
3% -55.27% 89.57% 0.000% 10.43%
The 2.5% CVaR of the 5 year cumulative geometric returns is displayed.
4.1 Portfolios Including Gold
In this section we will look at the results for a portfolio consisting of gold, stocks, government bonds and corporate bonds. The results for minimizing the standard deviation are in tables 24 and 25. The results including the corona crisis show that the minimum variance portfolio does not contain any gold. However when we increase our minimum return requirement we see an increase in the weights of gold. This is mainly due to the higher return of gold compared to bonds. The results excluding the corona crisis show similar weights in gold. However, we do see a small weight of gold in the minimum variance portfolio. The weight is very small so it does not significantly improve the standard deviation. However, it does show that the risk of gold is low enough to be considered in the minimum variance portfolio.
Table 24: Results minimizing standard deviation including corona
Minimum yearly return Std.dev Gold Stocks Gov-Bonds Corp-Bonds
None 3.767% 0.000% 0.000% 53.08% 46.92%
0% 3.785% 1.226% 0.498% 44.95% 53.32%
1% 6.807% 25.46% 13.40% 0.000% 61.14%
2% 13.58% 56.70% 32.18% 0.000% 11.12%
3% 26.54% 18.71% 81.29% 0.000% 0.000%
Table 25: Results minimizing standard deviation excluding corona
Minimum yearly return Std.dev Gold Stocks Gov-Bonds Corp-Bonds
None 2.605% 0.299% 0.000% 0.000% 99.70%
0% 2.605% 0.299% 0.000% 0.000% 99.70%
1% 4.182% 20.63% 15.54% 0.000% 63.83%
2% 7.965% 46.43% 36.73% 0.000% 16.82%
3% 14.16% 18.71% 81.29% 0.000% 0.000%
maximum CVaR portfolio. This is because increasing the return of an asset will shift its entire return distribution up.
Table 26: Results maximizing CVaR including corona
Minimum yearly return 2.5% CVaR Gold Stocks Gov-Bonds Corp-Bonds
None -21.38% 5.219% 0.436% 63.05% 31.30%
0% -21.46% 6.276% 1.166% 58.98% 33.58%
1% -32.21% 30.40% 11.21% 0.000% 58.39%
2% -52.24% 67.21% 27.52% 0.000% 5.264%
3% -80.25% 18.71% 81.29% 0.000% 0.000%
Table 27: Results maximizing CVaR excluding corona
Minimum yearly return 2.5% CVaR Gold Stocks Gov-Bonds Corp-Bonds
None -12.21% 4.911% 0.978% 0.000% 94.10%
0% -12.21% 4.911% 0.978% 0.000% 94.10%
1% -16.75% 23.80% 14.13% 0.000% 62.07%
2% -29.66% 54.89% 32.98% 0.000% 12.13%
3% -51.06% 18.71% 81.29% 0.000% 0.000%
Table 28: Weights of Gold for different Gold return assumptions
with corona without corona
Min return 0% 0.7% 1.6% 2.5% 0% 0.7% 1.6% 2.5%
None 2.834% 3.968% 5.219% 6.981% 0.000% 2.035% 4.911% 7.204%
0% 2.338% 5.336% 6.276% 6.679% 0.000% 2.035% 4.911% 7.204%
1% 0.000% 26.81% 30.40% 26.51% 0.000% 18.83% 23.80% 21.47%
2% 0.000% 50.15% 67.21% 55.77% 0.000% 43.10% 54.89% 49.57%
Lastly, we can also look at the improvement of the 2.5% CVaR for different return assumptions and the Benchmark case given earlier. The results are shown in tables 29 and 30. For both the case including and excluding the corona crisis we find that including gold to the benchmark portfolio slightly improves the CVaR of the portfolio. We especially see a large improvement to the portfolios with a higher minimum return restriction. The improvement is largest when we assume an expected return of 1.6% or 2.5% for gold. This shows that the size of the improvement is highly dependent on the return assumptions we use.
Table 29: CVaR’s for multiple Gold return assumptions including corona crisis.
Minimum yearly return Benchmark 0% 0.7% 1.6% 2.5%
None -21.72% -21.64% -21.54% -21.38% -21.17%
0% -22.19% -22.13% -21.84% -21.46% -21.18%
1% -42.94% -42.94% -39.70% -32.21% -27.73%
2% -68.78% -68.78% -65.11% -52.24% -41.56%
Table 30: CVaR’s for multiple Gold return assumptions excluding corona crisis.
Minimum yearly return Benchmark 0% 0.7% 1.6% 2.5%
None -12.37% -12.37% -12.35% -12.21% -11.97%
0% -12.37% -12.37% -12.35% -12.21% -11.97%
1% -22.05% -22.05% -20.52% -16.75% -14.05%
2% -40.65% -40.65% -37.84% -29.66% -22.73%
4.2 Portfolios Including Bitcoin
Table 31: Results minimizing standard deviation including corona crisis. Min yearly return Std.dev Bitcoin Stocks Gov-Bonds Corp-Bonds
None 3.792% 0.000% 0.689% 43.25% 56.06%
0% 3.792% 0.000% 0.689% 43.25% 56.06%
1% 8.404% 1.095% 22.99% 0.000% 75.91%
2% 17.77% 2.656% 53.21% 0.000% 44.13%
3% 28.37% 4.203% 83.13% 0.000% 12.66%
Table 32: Results minimizing standard deviation excluding corona crisis. Min yearly return Std.dev Bitcoin Stocks Gov-Bonds Corp-Bonds
None 2.605% 0.000% 0.000% 0.000% 100.00%
0% 2.605% 0.000% 0.000% 0.000% 100.00%
1% 4.899% 0.594% 23.76% 0.000% 75.65%
2% 9.889% 1.528% 54.94% 0.000% 43.53%
3% 15.37% 2.452% 85.81% 0.000% 11.73%
The results of optimizing the 2.5% CVaR are shown in tables 33 and 34. The results show that the weight of Bitcoin is slightly higher when we use CVaR as our risk measure. The difference is largest when we include the corona crisis and when we require a high minimum return on our portfolio. However, Bitcoin is still not a strong contender with stocks when we choose CVaR as our risk measure.
Table 33: Results maximizing CVaR including corona crisis.
Min yearly return 2.5% CVaR Bitcoin Stocks Gov-Bonds Corp-Bonds
None -21.72% 0.000% 0.737% 67.91% 31.35%
0% -22.11% 0.320% 2.623% 54.96% 42.09%
1% -40.55% 2.628% 20.65% 0.000% 76.71%
2% -65.30% 6.248% 47.71% 0.000% 46.04%
3% -80.55% 9.746% 74.64% 0.000% 15.62%
Table 34: Results maximizing CVaR excluding corona crisis.
Min yearly return 2.5% CVaR Bitcoin Stocks Gov-Bonds Corp-Bonds
None -12.37% 0.000% 0.595% 0.000% 99.41%
0% -12.37% 0.000% 0.595% 0.000% 99.41%
1% -21.51% 1.197% 22.84% 0.000% 75.97%
2% -39.49% 3.047% 52.61% 0.000% 44.34%
3% -53.77% 4.897% 82.07% 0.000% 13.03%
Table 35: Weights for different Bitcoin return assumptions
with corona without corona
Min return 0% 5% 10% 0% 5% 10% none 0.000% 0.000% 0.216% 0.000% 0.000% 0.009% 0% 0.000% 0.320% 0.557% 0.000% 0.000% 0.009% 1% 0.000% 2.628% 3.481% 0.000% 1.197% 2.070% 2% 0.000% 6.248% 7.916% 0.000% 3.047% 4.669% 3% 0.000% 9.746% 12.15% 0.000% 4.897% 7.258%
We now investigate whether adding this small weight of Bitcoin to our benchmark portfolio improves the 2.5% CVaR. The results of the 2.5% CVaR for the different Bitcoin return assumptions and the benchmark portfolio are given in table 36. The results show no improvements when we assume a 0% return, since the weight of Bitcoin is zero in that case. For a 5% return assumption we see only very small and insignificant improvements. For the 10% return assumption we see a somewhat larger improvement, however this is still a small amount when we compare it to the improvements made by gold.
Table 36: CVaR’s for different Bitcoin return assumptions
with corona without corona
Min return Benchmark 0% 5% 10% Benchmark 0% 5% 10%
none -21.72% -21.72% -21.72% -21.70% -12.37% -12.37% -12.37% -12.37%
0% -22.19% -22.19% -22.11% -21.90% -12.37% -12.37% -12.37% -12.37%
1% -42.93% -42.93% -40.55% -36.33% -22.05% -22.05% -21.51% -19.88%
2% -68.78% -68.78% -65.30% -58.01% -40.65% -40.65% -39.49% -36.18%
3% -83.41% -83.41% -80.55% -73.54% -55.27% -55.27% -53.77% -49.67%
4.3 Portfolios Including Gold and Bitcoin
In this section we will look at the effect of adding both gold and Bitcoin to our benchmark portfolio. For this analysis we have chosen the standard return assumption of 1.6% for gold and we will be using the 0%, 5% and 10% return assumptions for Bitcoin. Like in the previous section we first show the complete results assuming a 5% return on Bitcoin and after that show the results for different returns on Bitcoin. The results for the Mean-Variance optimization are given in tables 37 and 38. We find that the weight of Bitcoin is significantly smaller when we also add gold to the portfolio. Therefore, some of the improvements made by Bitcoin to our benchmark portfolio are mitigated when we also include gold to this benchmark portfolio. Furthermore, we can see that the weight of gold stays almost the same. Therefore, adding Bitcoin to the benchmark portfolio does not significantly impact the improvements gold can make to the portfolio.
Table 37: Results minimizing standard deviation including corona crisis. Min yearly return Std.dev Bitcoin Gold Stocks Gov-Bonds Corp-Bonds
none 3.767% 0.000% 0.000% 0.000% 53.08% 46.92%
0% 3.785% 0.000% 1.226% 0.498% 44.95% 53.32%
1% 6.746% 0.576% 24.72% 12.84% 0.000% 61.86%
2% 13.35% 1.508% 54.76% 30.73% 0.000% 13.01%
Table 38: Results minimizing standard deviation excluding corona crisis. Min yearly return Std.dev Bitcoin Gold Stocks Gov-Bonds Corp-Bonds
none 2.604% 0.000% 0.299% 0.000% 0.000% 99.67%
0% 2.604% 0.000% 0.299% 0.000% 0.000% 99.67%
1% 4.164% 0.365% 20.33% 15.11% 0.000% 64.17%
2% 7.885% 1.014% 45.59% 35.54% 0.000% 17.85%
3% 13.75% 2.842% 21.43% 75.73% 0.000% 0.00%
The results for maximizing the 2.5% CVaR are shown in tables 39 and 40. We can see that the weight of Bitcoin is lower for all the minimum return restrictions. for the 1 and 2% the difference is quite large, however for the 3% minimum return restriction the difference is negligible. The weight of gold is mainly subtracted from the weight in stocks. For gold we see no large differences when we add Bitcoin to the portfolio for all the minimum return requirements except for the 3% level. For the 3% level we actually see a significantly higher weight of gold in the portfolio when we include Bitcoin. The weight of stocks goes down from approximately 81% to 62% compared to the case including only gold. The 19% difference in stocks is almost 50/50 reallocated into Bitcoin and gold. This 50/50 allocation in Bitcoin and gold gives approximately the same return as stocks and apparently is less risky. However, these results are very dependent on the return assumptions of all our assets.
Table 39: Results maximizing CVaR including corona crisis.
Min yearly return 2.5% CVaR Bitcoin Gold Stocks Gov-Bonds Corp-Bonds
none -21.38% 0.083% 5.275% 0.455% 63.08% 31.10%
0% -21.44% 0.204% 5.982% 1.117% 59.63% 33.07%
1% -31.73% 0.893% 29.29% 10.33% 0.000% 59.48%
2% -50.71% 2.688% 62.70% 25.40% 0.000% 9.21%
3% -75.57% 9.660% 27.94% 62.40% 0.000% 0.000%
Table 40: Results maximizing CVaR excluding corona crisis.
Min yearly return 2.5% CVaR Bitcoin Gold Stocks Gov-Bonds Corp-Bonds
none -12.21% 0.000% 4.911% 0.978% 0.000% 94.10%
0% -12.21% 0.000% 4.911% 0.978% 0.000% 94.10%
1% -16.67% 0.535% 23.25% 13.56% 0.000% 62.66%
2% -29.17% 1.608% 53.10% 31.31% 0.000% 13.98%
3% -48.57% 4.912% 23.40% 71.68% 0.000% 0.00%
