在衡量投資組合的風險時，風險值是一種常用的方式。在本文中，我們採用copula-GARCH和其他四種傳統方法來看哪一種方法在估計比特幣投資組合的風險值時表現更好，本文以黃金和以太幣分別與比特幣建立投資組合。在copula-GARCH模型中，我們選擇了四個靜態和兩個動態copula模型，結合殘差為Student-t分配的GARCH(1,1)模型，以建構兩個資產的聯合分佈。另外，本文採用的傳統方法為歷史模擬法、變異數共變異數、指數加權移動平均法和GARCH風險值法，我們採用了每種方法來計算一日風險值。本文實證結果顯示，對於由比特幣和以太幣組成的投資組合，copula-GARCH模型在估算風險值的表現優於其餘傳統方法；而對於比特幣和黃金組成的投資組合，傳統方法在估算風險值時表現較好。我們推測可能的原因是，由於比特幣報酬率和黃金報酬率之間的相關性非常低，copula-GARCH模型可能在相關性低的情況下表現不佳。

When it comes to measuring risk of a portfolio, value-at-risk (VaR) is a commonly used way. In this paper, we applied the copula-GARCH method and other traditional methods to determine which one is better for estimating the VaR of portfolios containing Bitcoin. Gold and Ethereum were used to construct a portfolio with Bitcoin respectively. For copula-GARCH model, we selected four constant and two time-varying copula models combined with GARCH Student-t residuals to fit the joint distribution of the two assets in the portfolios. The traditional methods are referred to historical simulation, variance-covariance method, EWMA method and univariate-GARCH VaR method. We adopted each method to compute corresponding one-day VaRs. Our results indicated that, for the portfolio contained Bitcoin and Ethereum, copula-GARCH performed better than traditional methods; while for the portfolio consisted of Bitcoin and gold, traditional methods performed better. The correlation coefficient between Bitcoin and gold is extremely low. Hence, the results inferred that copula-GARCH may not be suitable in the extremely low correlation case.

1 Introduction………………………………………………………………………………………1
2 Methodology…………………………………………………………………………………………3
2.1 Value at Risk……………………………………………………………………………………3
2.1.1 Historical Simulation………………………………………………………………4
2.1.2 Variance-Covariance Method…………………………………………………4
2.1.3 Exponentially Weighted Moving Average……………………4
2.1.4 Univariate GARCH-VaR…………………………………………………………………5
2.2 Copula-GARCH Model………………………………………………………………………5
2.2.1 Sklar’s Theorem………………………………………………………………………………5
2.2.2 Copula Functions……………………………………………………………………………6
2.2.3 Parameter Estimation…………………………………………………………………8
2.2.4 VaR using Copula-GARCH……………………………………………………………9
2.2.5 Back Testing………………………………………………………………………………………9
3 Data…………………………………………………………………………………………………………10
3.1 Portfolio containing Bitcoin and Gold…………………10
3.2 Portfolio containing Bitcoin and Ethereum………10
4 Empirical Results………………………………………………………………………11
4.1 Marginal Distributions…………………………………………………………11
4.2 Copulas…………………………………………………………………………………………………11
4.3 Backtesting of VaR using Five Methods…………………11
5 Conclusion…………………………………………………………………………………………13
References………………………………………………………………………………………………………………14
List of Tables and Figures……………………………………………………………………16

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