Cryptocurrency portfolio optimization with multivariate normal tempered stable processes and Foster-Hart risk
CCryptocurrency portfolio optimization with multivariatenormal tempered stable processes and Foster-Hart risk ∗ Tetsuo Kurosaki † , Young Shin Kim ‡ October 20, 2020
Abstract
We study portfolio optimization of four major cryptocurrencies. Our time seriesmodel is a generalized autoregressive conditional heteroscedasticity (GARCH) modelwith multivariate normal tempered stable (MNTS) distributed residuals used to cap-ture the non-Gaussian cryptocurrency return dynamics. Based on the time seriesmodel, we optimize the portfolio in terms of Foster-Hart risk. Those sophisticatedtechniques are not yet documented in the context of cryptocurrency. Statistical testssuggest that the MNTS distributed GARCH model fits better with cryptocurrency re-turns than the competing GARCH-type models. We find that Foster-Hart optimizationyields a more profitable portfolio with better risk-return balance than the prevailingapproach.
Keywords:
Cryptocurrencies, Foster-Hart risk, GARCH modeling, Multivariate nor-mal tempered stable process, Portfolio optimization, Value at risk
JEL Classification Numbers:
C13, C22, C52, C61, G11 ∗ We are grateful to Professor Svetlozar Rachev, in Texas Tech University, for his continuous guidance andencouragements on the research topic of this paper. For the first author, the opinions, findings, conclusionsor recommendations expressed in this paper are our own and do not necessarily reflect the views of theBank of Japan. † Bank of Japan, 2-1-1 Nihonbashi-Hongokucho, Chuo-ku, Tokyo 103-8660, Japan;E-mail address: [email protected] ‡ College of Business, Stony Brook University, Stony Brook, NY 11794-3775, United States;E-mail address: [email protected] a r X i v : . [ q -f i n . P M ] O c t Introduction
Cryptocurrency is an entirely new finanical asset, which increases market capitalizationrapidly and attracts growing attention from market participants. The advantages of cryp-tocurrencies over traditional currencies are abundant, and include reliability and anonymityin transactions with lower costs that are facilitated by novel blockchain technology. De-spite the appeal of these advantages, cryptocurrencies present potential vulnerabilities inthat they do not enjoy the aegis of central banks or any other monetary authorities. Thisleads to price volatility in the face of real economic events, such as the recent Covid-19pandemic, ultimately strengthening global calls for regulations on cryptocurrency trading. These peculiar price movements motivate scholars to study the impact of cryptocurrencieson financial markets.From an econometric point of view, it is of interest to explore which kind of timeseries model accounts for highly volatile returns on and accurately forecasts risks asso-ciated with cryptocurrencies. Among expanding literature, the studies by Caporale andZekokh (2019), Cerqueti et al. (2020), and Troster et al. (2019) are based on a general-ized autoregressive conditional heteroscedasticity (GARCH) model. They share the com-mon conclusion that the normally distributed GARCH model is inadequate for describingcryptocurrency returns and the introduction of non-Gaussian distribution substantiallyimproves the goodness-of-fit of a GARCH-type model. Another approach includes thestochastic volatility model (Chaim and Laurini, 2018), and the generalized autoregressivescore model (Troster et al., 2019). Also, the high volatility of cryptocurrency highlights itsspeculative nature. Brauneis and Mestel (2019) investigate the risk-return relationship ofan optimized cryptocurrency portfolio based on the Markowitz mean-variance framework.This paper studies the portfolio optimization of cryptocurrencies by employing a unionof sophisticated time series models and risk measures. Four major cryptocurrencies areselected as samples. Our time series model is the multivariate normal tempered stable(MNTS) distributed GARCH model. The MNTS distribution (Kim et al., 2012) hasdemonstrated excellent fit to joint dynamics of physical asset returns in a number of em-pirical studies (Anand et al., 2016; Anand et al., 2017; Bianchi and Tassinari, 2020; Kimet al., 2015; Kurosaki and Kim, 2013a; Kurosaki and Kim, 2013b; Kurosaki and Kim, 2019;and Shao et al., 2015). Our portfolio optimization strategy is based on Foster-Hart risk,which is very sensitive to risky left tail events. Foster-Hart (FH, hereafter) risk was origi-nally introduced in the field of game theory (Foster and Hart, 2009), and was subsequentlyapplied to risk management related to financial markets (Anand et al., 2016; Kurosaki andKim, 2019; Leiss and Nax, 2018). These cutting-edge techniques are not yet documented Basel Committee on Banking Supervision (2019) emphasizses that cryptocurrency is not legal tenderand warns about the potential financial stability concerns caused by its continuous growth.
2n the context of cryptocurrency. Statistical tests demonstrate that the MNTS distributedGARCH model has better explanatory power for cryptocurrency returns than the normallydistributed GARCH model. Also, we find that the model creates more profitable portfoliosin tandem with FH risk than the traditional mean-variance approach.The rest of this paper is organized as follows. Section 2 briefly introduces our method-ology. Section 3 describes the data of cryptocurrency used herein. Section 4 outlines theempirical results of statistical tests for each cryptocurrency return. Section 5 conductsportfolio optimization and discusses performance. Section 6 summarizes our findings.
We introduce the non-Gaussian time series model with the MNTS distribution as wellas FH risk, which work together to achieve efficient portfolio optimization (Anand et al.,2016). See also A for supplementary explanations.
We utilize a GARCH-type model to describe the dynamics of cryptocurrency returns.Given that both autoregressive (AR) and moving average (MA) processes are typicallyobserved in financial data, we apply the most standard ARMA(1,1)-GARCH(1,1) specifi-cation. After GARCH filtering, we obtain independent and identically distributed (i.i.d.)standard residuals η t with a mean of zero and unit variance for each cryptocurrency. Inorder to describe complicated interdependency among cryptocurrencies, we conduct multi-variate modeling on each cryptocurrency’s η t jointly. We employ an i.i.d. standard MNTSas an assumptive distribution that η t follows. We also hypothesize that η t follows an i.i.d.standard normal and student t as competing models. Hereafter, we denote the ARMA(1,1)-GARCH(1,1) model with multivariate normal, student t, and NTS distributed standardresiduals as AGNormal, AGT, and AGNTS, respectively.We prefer the MNTS to other miscellaneous non-Gaussian distributions because of itsflexibility with respect to a multivariate extension. Both the estimation of the MNTS fromreal data and the scenario generation based on the estimated MNTS are feasible withoutcomputational difficulty even in considerably high dimensional settings. These features arecritical in their application to portfolio optimization. Also, the reproductive property ofthe stable distribution has an affinity for portfolio modeling. We introduce FH risk. Let a gamble be any bounded random variable g with a positiveexpected value and a positive probability of losses: E ( g ) > P ( g < >
0. FH risk is3he minimum reserve that an agent should initially possess to prevent himself from almostcertainly going bankrupt, even after the infinite repetition of the gamble g . Foster andHart (2009) demonstrate that, for a gamble g , irrespective of the utility function, FH risk R ( g ) is the unique positive root of the following equation: E (cid:18) log (cid:20) gR ( g ) (cid:21)(cid:19) = 0 . (1)The bankruptcy-proof property endows FH risk with extremely high sensitivity to negativeevents. By regarding investments in financial assets as a gamble, FH risk is expected tosense market downturn in a forward-looking manner.We also utilize more popular risk measures, Value at Risk (VaR) and Average VaR(AVaR), in order to supplement FH risk. Accuracy of risk forecasting is an importantaspect of time series models. Statistically backtesting VaR and AVaR is feasible due totheir relative simplicity, whereas no backtesting methodology has been established for FHrisk. We backtest VaR by the Christoffersen’s likelihood ratio (CLR) test as well as AVaRby the Berkowitz’s likelihood ratio (BLR) tail test and Acerbi and Szekely (AS) test. SeeChristoffersen (1998), Berkowitz (2001), and Acerbi and Szekely (2014), respectively.
Our dataset contains daily logarithmic returns of cryptocurrency exchange spot rates inU.S. Dollars per unit from 2015/08/31 to 2020/03/31, resulting in 1,674 observations foreach cryptocurrency. Following Caporale and Zekokh (2019), we select the following fourcryptocurrencies as samples: Bitcoin (BTC), Ethereum (ETH), Litecoin (LTC), and XRP.The data source is CoinMarketCap. Table 1 reports the descriptive statistics of ourdataset. All cryptocurrencies have larger kurtosis than those of the normal distribution,and show either negative or positive skewness. These observations motivate us to apply thenon-Gaussian model. We estimate all models based on maximum likelihood estimation.In order to obtain the AGNTS model, we estimate the univariate AGT model for eachcryptocurrency first and subsequently fit the standard MNTS distribution to the sameresiduals η t . We refer the readers to Kurosaki and Kim (2019) for details regarding thesetechniques.Our analysis procedure is as follows. First, we arrange a moving window with a lengthof 500 days. The first window, ranging from 2015/08/31 to 2017/01/12, moves aheadday by day until 2020/03/31, which amounts to 1,175 distinct windows. Subsequently, we AVaR is also called Conditional VaR or Expected Shortfall. https://coinmarketcap.com/ We assess the capability of our multivariate GARCH-type models to account for marginalreturn dynamics of each cryptocurrency. Specifically, we examine the statistical perfor-mance of the 1,175 iteratively-estimated models based both on in-sample and out-of-sampletests.
As an in-sample test, we investigate the fitting performance of standard residuals η t ofthe univariate ARMA(1,1)-GARCH(1,1) model for the assumptive distributions (normal,student t, and NTS). To do so, we exploit both the Kolmogorov-Smirnov (KS) and theAnderson-Darling (AD) tests. While both tests are designed to assess the goodness-of-fitof the proposed distributions, the AD test puts more emphasis on fitting at the tail. Underthe reasonable postulation that our sample is sufficient in number, we can compute p-valuesfor both tests.Tables 2 and 3 report the number of rejections of KS and AD tests out of 1,175 iteratedestimations for AGNormal, AGT, and AGNTS residuals, respectively, by significance level.We see that AGNormal is almost always rejected by both tests and thus significantly un-derperforms AGT and AGNTS. In the KS test, AGNTS has a smaller number of rejectionsthan AGT in three out of four cryptocurrencies at the 10% level. More clearly, in theAD test, AGNTS has a smaller number of rejections than AGT in four (three) out of fourcryptocurrencies at the 5% (10%) level due to the excellent ability of AGNTS to track tailbehavior. Overall, AGNTS is the most preferable model. As an out-of-sample test, we backtest risk measures, namely, VaR and AVaR. Each ofthe iteratively-estimated models forecasts one-day-ahead VaR and AVaR, constituting atime series of VaR and AVaR forecasts with a length of 1,175 days from 2017/01/12 to2020/03/31. In line with the Basel accord, we adopt the 99% confidence level for VaRand AVaR. Backtesting is achieved by comparing VaR and AVaR with actual returns every Following Kim et al. (2010), the VaR estimation with AGNTS relies on the discrete Fourier Transform. First of all, AGNormal has lower p-values than AGT and AGNTS, especially in the BLRand AS tests. AGNTS passes the CLR tests during any period and with any cryptocur-rency, including at the 10% level, whereas AGT fails in Periods 1 and 2 at the same level.Also, AGNTS always passes the BLR tests except for in Period 2 and in BTC. By contrast,AGT fails in Periods 1 and 2 at the 5% level. Finally, AGNTS fails the AS tests for atmost one cryptocurrency in each subperiod at the 10% level. However, AGT fails for twocryptocurrencies in Periods 1 and 3 at the same level. Therefore, we conclude that AGNTSshows the best performance in out-of-sample tests more clearly than in in-sample tests.
We practice portfolio optimization with cryptocurrencies consisting of BTC, ETH, LTC,and XPR, in line with Kurosaki and Kim (2019). The portfolio risk and reward is forecastedthrough multivariate time series models. The optimization is carried out by minimizingthe objective risk measure under the tradeoff with expected returns and transaction costsfollowing the procedure detailed in B. Since, as is shown in Section 4, AGNTS provides amore accurate expression than AGNormal and AGT for leptokurtic and skewed behaviorsof cryptocurrency returns, it is also expected to show better performance in portfoliooptimization. We exploit FH risk as the objective risk measure to be minimized, as wellas standard deviation (SD) and AVaR.Table 5 exhibits optimization results under the absence of transaction costs ( λ = 0),through a combination of time series models and objective risk measures. When the port-folio is optimized with respect to SD, AVaR, and FH under the tradeoff against expectedreturns, we refer to the corresponding portfolio as mean-SD, mean-AVaR, and mean-FHportfolio, respectively. Column 2 reports the cumulative returns that each optimized port- The p-values of the AS test are computed from 10 sample statistics generated by time series modelsand for a left tail. We see that the mean-FHportfolio with AGNTS forecasts yields the largest profit, followed by the mean-AVaR port-folio with AGNTS forecasts. Columns 3 through 5 show the SD, AVaR, and FH risk ofthe optimized portfolio itself, which are computed from their historical returns. Columns6 through 8 indicate the cumulative returns adjusted by each risk measure. Note thatthe return-to-SD ratio in Column 6 is the well-known Sharpe ratio. We find that themean-FH portfolio with AGNTS forecasts shows the highest return-to-risk ratios of anyrisk measures. Therefore, we conclude that this combination not only achieves the largestcumulative returns but also generates the most ideal balance between risk and reward.As a robustness check, we conduct the optimization in the presence of transaction costs,where the corresponding cost ( λ · | w t − w t − | ) is deducted from daily returns. Figure 1shows how the cumulative returns of the optimized portfolio evolve temporarily over theinvestment period based on different investor cost aversion ( C = 0 . , . , λ = 0). We focus on the forecasts of future returns createdby AGNTS. We observe that the mean-FH portfolio accrues the largest profit irrespectiveof cost aversion parameters. Therefore, the superiority of the combination of AGNTS andFH risk still holds for the case where the transaction cost is present. This paper studies the portfolio optimization of four major cryptocurrencies. A cryp-tocurrency as an asset class is characterized by higher volatility and more nonlinear returndynamics compared to traditional assets. Statistical analysis demonstrates that the intro-duction of the MNTS distribution enhances the explanatory power of the GARCH-typemodel for cryptocurrency return dynamics substantially, especially in terms of risk fore-casting. FH risk warns of market crashes, for example, in scenarios such as those causedby the Covid-19 pandemic, in an immeasurably sensitive manner. The combination of theMNTS distributed GARCH model and FH risk leads to desirable portfolio optimizationwith respect to cumulative returns as well as risk-return balance. We first document theeffectiveness of those sophisticated techniques in the context of cryptocurrency. Since AGT and AGNTS share the same residuals, both models produce the same mean-SD portfolio.Also, note that the first revenue recognition takes place on the day after the first optimization, 2017/01/13. The presented optimization allows for both long and short positions. As a further robustness check,we conduct the optimization under the restriction that a short position is prohibited. This restrictionis realistic for conservative investors. The results are consistent with unrestricted cases; The mean-FHportfolio and/or the AGNTS forecasts generally create the largest profit. eferences Acerbi, C., Szekely, B., Dec. 2014. Backtesting expected shortfall. Working paper, MSCI.URL
Anand, A., Li, T., Kurosaki, T., Kim, Y. S., 2016. Foster-Hart optimal portfolios. Journalof Banking & Finance 68, 117–130.Anand, A., Li, T., Kurosaki, T., Kim, Y. S., 2017. The equity risk posed by the too-big-to-fail banks: A Foster-Hart estimation. Annals of Operations Research 253, 21–41.Basel Committee on Banking Supervision, 2019. Statement on crypto-assets. Newsletters,Bank for International Settlements.Berkowitz, J., 2001. Testing density forecasts, with applications to risk management. Jour-nal of Business & Economic Statistics 19, 465–474.Bianchi, M. L., Tassinari, G. L., 2020. Forward-looking portfolio selection with multivariatenon-gaussian models. Quantitative Finance.Brauneis, A., Mestel, R., 2019. Cryptocurrency-portfolios in a mean-variance framework.Finance Research Letters 28, 259–264.Caporale, G. M., Zekokh, T., 2019. Modelling volatility of cryptocurrencies using Markov-Switching GARCH models. Research in International Business and Finance 48, 143–155.Cerqueti, R., Giacalone, M., Mattera, R., 2020. Skewed non-gaussian GARCH models forcryptocurrencies volatility modelling. Information Sciences 527, 1–26.Chaim, P., Laurini, M. P., 2018. Volatility and return jumps in bitcoin. Economics Letters173, 158–163.Christoffersen, P. F., 1998. Evaluating interval forecasts. International Economic Review39, 841–862.Fabozzi, F. J., Focardi, S. M., Kolm, P. N., 2006. Financial Modeling of the Equity Market:From CAPM to Cointegration. John Wiley & Sons, Inc., New Jersey.Foster, D. P., Hart, S., 2009. An operational measure of riskiness. Journal of PoliticalEconomy 117 (5), 785–814.Kim, Y. S., Giacometti, R., Rachev, S. T., Fabozzi, F. J., Mignacca, D., 2012. Measuringfinancial risk and portfolio optimization with a non-Gaussian multivariate model. Annalsof Operations Research 201, 325–343. 8im, Y. S., Lee, J., Mittnik, S., Park, J., 2015. Quanto option pricing in the presence offat tails and asymmetric dependence. Journal of Econometrics 187 (2), 512–520.Kim, Y. S., Rachev, S. T., Bianchi, M. L., Fabozzi, F. J., 2010. Computing VaR and AVaRin infinitely divisible distributions. Probability and Mathematical Statistics 30, 223–245.Kurosaki, T., Kim, Y. S., 2013a. Mean-CoAVaR optimization for global banking portfolios.Investment Management and Financial Innovation 10 (2), 15–20.Kurosaki, T., Kim, Y. S., 2013b. Systematic risk measurement in the global banking stockmarket with time series analysis and CoVaR. Investment Management and FinancialInnovation 10 (1), 184–196.Kurosaki, T., Kim, Y. S., 2019. Foster-Hart optimization for currency portfolios. Studiesin Nonlinear Dynamics & Econometrics 23 (2), 20170119.Leiss, M., Nax, H. H., 2018. Option-implied objective measures of market risk. Journal ofBanking & Finance 88, 241–249.Shao, B. P., Rachev, S. T., Mu, Y., 2015. Applied mean-ETL optimization in using earningsforecasts. International Journal of Forecasting 31 (2), 561–567.Troster, V., Tiwari, A. K., Shahbaz, M., Macedo, D. N., 2019. Bitcoin returns and risk: Ageneral GARCH and GAS analysis. Finance Research Letters 30, 187–193.9able 1: Descriptive statistics of our dataset. The sample period is from 2015/08/31 to2020/03/31.
Cryptocurrency Number ofobservation Mean Max Min Standarddeviation Kurtosis SkewnessBTC 1674 0.0020 0.2251 − . − . − . − . − . − . Table 2: Number of rejections of Kolmogorov-Smirnov tests out of 1,175 iterated estima-tions for GARCH residuals
Significancelevel 1% 5% 10%Model AGNormal AGT AGNTS AGNormal AGT AGNTS AGNormal AGT AGNTSBTC 1009 92 107 1079 186 185 1163 303 259ETH 1110 136 134 1173 337 383 1175 542 523LTC 889 416 488 1167 532 577 1175 648 693XRP 1174 683 650 1175 783 764 1175 812 791
Table 3: Number of rejections of Anderson-Darling tests out of 1,175 iterated estimationsfor GARCH residuals
Significancelevel 1% 5% 10%Model AGNormal AGT AGNTS AGNormal AGT AGNTS AGNormal AGT AGNTSBTC 1022 137 81 1175 537 155 1175 629 236ETH 1142 106 103 1175 329 287 1175 503 523LTC 1144 377 462 1175 544 543 1175 665 648XRP 1175 557 729 1175 812 801 1175 835 821
Test CLR BLR ASModel AGNormal AGT AGNTS AGNormal AGT AGNTS AGNormal AGT AGNTSPeriod 1 (2017/01/12 to 2018/03/31)BTC 0.0020 0.0174 0.7307 0.0003 0.0338 0.7824 0.0000 0.0186 0.6921ETH 0.0450 0.6691 0.6691 0.0000 0.6304 0.8778 0.0000 0.5016 0.3827LTC 0.4489 0.4489 0.4489 0.0009 0.7565 0.7650 0.5963 0.8748 0.8455XRP 0.0450 0.2217 0.4153 0.0000 0.5008 0.5698 0.0000 0.0835 0.0510Number of p-valuesless than 5% 3 1 0 4 1 0 3 1 0Number of p-valuesless than 10% 3 1 0 4 1 0 3 2 1Period 2 (2018/04/01 to 2019/03/31)BTC 0.0388 0.0990 0.2236 0.0000 0.1242 0.0546 0.0000 0.0014 0.0001ETH 0.0137 0.4419 0.7271 0.0000 0.3352 0.8721 0.0000 0.1670 0.2896LTC 0.4419 0.2236 0.4419 0.0872 0.0064 0.1640 0.0319 0.2083 0.2491XRP 0.7271 0.3366 0.3366 0.5057 0.3174 0.5176 0.2599 0.9671 0.9657Number of p-valuesless than 5% 2 0 0 2 1 0 3 1 1Number of p-valuesless than 10% 2 1 0 3 1 1 3 1 1Period 3 (2019/04/01 to 2020/03/31)BTC 0.0394 0.4449 0.7295 0.0000 0.4436 0.2497 0.0000 0.0177 0.0420ETH 0.1002 0.7295 0.6733 0.0000 0.9374 0.8386 0.0000 0.3157 0.6138LTC 0.1002 0.4449 0.7295 0.0000 0.9158 0.9124 0.0000 0.1602 0.2592XRP 0.2258 0.2258 0.4449 0.0000 0.5051 0.7859 0.0000 0.0731 0.1274Number of p-valuesless than 5% 1 0 0 4 0 0 4 1 1Number of p-valuesless than 10% 1 0 0 4 0 0 4 2 1 λ = 0) Portfolio CumulativeReturn (a) SD (b) AVaR (c) FH risk (d) a/b a/c a/dAGNormalMean-SD 0.8662 0.0415 0.2293 1.2554 20.8968 3.7773 0.6900Mean-AVaR − . − . − . − . -1012 (a) λ = 0 -101234Jan-17 Jul-17 Jan-18 Jul-18 Jan-19 Jul-19 Jan-20Mean-SD Mean-AVaR Mean-FHCumulative return (b) C = 0 . -101234Jan-17 Jul-17 Jan-18 Jul-18 Jan-19 Jul-19 Jan-20 Mean-SD Mean-AVaR Mean-FHCumulative return (c) C = 0 . -101234Jan-17 Jul-17 Jan-18 Jul-18 Jan-19 Jul-19 Jan-20Mean-SD Mean-AVaR Mean-FHCumulative return (d) C = 1 Figure 1: Time Evolution of Optimized Portfolio’s Cumulative Returns.12
Supplement to Methodology
A.1 The MNTS distribution
A real parameter set ( α, θ, β, γ, µ,
Σ) characterizes the n -dimensional MNTS distribution,where α ∈ (0 , θ > β, µ ∈ R n , γ ∈ R n + , and Σ is a n -by- n correlation matrix (Kimet al., 2012). The common parameters ( α, θ ) control fat-tailness. Each component of β , γ ,and µ corresponds to each marginal; β controls skewness, γ i scales the distribution, and µ is a mean value. The correlation matrix Σ is responsible for asymmetric interdependenceamong marginals. These unique parameters enable the MNTS to have explanatory powersfor the stylized nature of financial asset returns, such as fat-tailness and skewness, whicha normal distribution fails to capture. See Kim et al. (2012) for more detail. A.2 VaR and AVaR
The VaR at the 1 − (cid:15) confidence level is the (cid:15) -quantile of an asset loss. AVaR is theexpected loss on the condition that it is at levels in excess of VaR. As well as FH risk, AVaRis a convex function with a unique minimum. Therefore, the portfolio can be effectivelyoptimized with respect to AVaR and FH risk. This is not the case with VaR because itgenerally has multiple local minima. It is also noteworthy that VaR and AVaR should bedefined with time horizon and confidence level, whereas FH risk is an absolute risk measureirrespective of such predetermined parameters.The Christoffersen’s likelihood ratio (CLR) test is a well-known method for backtest-ing VaR as an interval forecast (Christoffersen, 1998). More specifically, we basktest VaRthrough a CLR test with a conditional coverage property, which is the joint test of un-conditional coverage and independence, because it can take the tendency for consecutiveVaR breaches into consideration. Furthermore, we backtest AVaR both directly and indi-rectly. For the indirect approach, we utilize a Berkowitz’s likelihood ratio (BLR) tail test(Berkowitz, 2001). The BLR tail test can backtest the accuracy of tail behavior forecastsof a given distribution. Accurate tail behavior forecasts coincide with accurate AVaR fore-casts. For our direct approach, we exploit the Acerbi and Szekely (AS) test (Acerbi andSzekely, 2014). While they propose several statistics so as to backtest the estimated AVaRat the 1 − (cid:15) confidence level, we choose the following statistic: Z = T (cid:88) t =1 R t I t T (cid:15)
AVaR (cid:15),t + 1 , (2)where R t is a historical asset return, I t is the indicating function for VaR breaches, i.e., I t = 1 − R t > VaR (cid:15),t , T is the forecast length, and t is the time period. If the AVaR forecast is13ccurate (the null hypothesis), Z must be zero; otherwise, Z is negative. The p-values ofthe AS test can be derived by simulating R t through a time series model and sampling Z. B Portfolio Optimization Procedure
The presented problem is to search for the optimized weight of each cryptocurrency i withexpected return µ i,t at time t , denoted by w i,t (1 ≤ i ≤ t , under the tradeoff between risk and reward. We identify thisproblem as minimizing the risk-to-reward ratio (disutility), where the risk and the rewardare quantified by FH risk and expected returns, respectively. We supplementarily utilizeone-day-ahead AVaR at the 99% confidence level and standard deviation (SD) as riskmeasures for reference. We exclude VaR because of its non-convexity. Note that theoptimization based on SD is equivalent to the classical Markowitz framework.We impose some constraints on weights w i,t . First, we allow for both long and shortpositions up to unit, since short positions are typical in cryptocurrency trading especiallyafter the issuance of Bitcoin futures in December, 2017. Second, we maintain positiveexpected returns from the portfolio. The second constraint stems from the principle ofspeculation as well as the necessary condition for FH risk to be defined. In addition, weconsider the cost of reallocating portfolio weights, which partly deprives the portfolio ofcumulative returns. We also take investor cost aversion into account.Consequently, our optimization problem is described using the following equation:min − ≤ w i,t ≤ ≤ w (cid:62) t µ t ρ ( w t ) w (cid:62) t µ t + C (cid:20) λ · ( w t − w t − ) w (cid:62) t µ t (cid:21) , (3)where w t = ( w ,t , . . . , w ,t ) (cid:62) , µ t = ( µ ,t , . . . , µ ,t ) (cid:62) , and ρ ( · ) is a risk measure of the port-folio. The second quadratic term of the change in weights corresponds to the transactioncost (e.g., Fabozzi et al., 2006). λ and C are the parameters for transaction cost chargedper unit weight change and investor cost aversion relative to risk-to-reward ratio, respec-tively. We set λ as 10 − when transaction costs are present. We assume several distinctvalues for C since it depends on investors. Notice that our objective function (3) is set ashomogeneous with respect to the size of w t , as long as ρ ( · ) satisfies the homogeneity.We therefore have the 1,175 iteratively-estimated time series models from 2017/01/12to 2020/03/31. Every day during this investment period, we generate one-day-ahead 10 scenarios for each cryptocurrency return by utilizing the estimated AGNormal, AGT, andAGNTS models. Under these generated scenarios, we forecast risk and reward, thereby Since µ i,t is typically on the order of 10 − in our dataset, λ = 10 − suggests that transaction costs areroughly on the order of 1 bps to the portfolio expected returns. w t to solve equation (3). The portfolio is optimized into w t at the end of day t and create profit or loss from the return at the end of day tt