﻿ 基于GARCH-EVT-Copula的WCVaR鲁棒投资组合模型

# 基于GARCH-EVT-Copula的WCVaR鲁棒投资组合模型Robust Portfolio Selection with GARCH-EVT-Copula

Abstract: In this paper, a robust portfolio model is established under the assumption that the underlying distribution is mixed and uncertain. Firstly, we use the GARCH-EVT model to describe the fat tail and heteroscedasticity characteristics of single financial asset return. Then we use Copula to describe the dependence structure between the yields, and establish the GARCH-EVT-Copula model. Finally, a mixture of uncertain multivariate distributions of returns is constructed using different Copula functions, and a robust portfolio model of GARCH-EVT-Copula-WCVaR is estab-lished on the basis of WCVaR-Copula robust model. Compared to the Normal-Copula-WCVaR robust model and the classical mean variance model in the experimental study, the return of GARCH-EVT-Copula-WCVaR robust model is higher than that of Normal-Copula-WCVaR robust model and mean variance model during the stock market crisis and the period when the volatility of return fluctuates.

1. 引言

1952年Markowitz提出的均值方差模型奠定了现代投资组合理论的基础，在资产配置领域取得了广泛的应用。但是，该模型存在对参数估计非常敏感的不足，Black与Litterman (1992) [1] 指出在使用均值方差投资组合模型时，投资组合策略对收益率均值十分敏感。为了降低投资组合模型对参数估计的敏感性，鲁棒优化被引入到投资组合理论中。鲁棒优化是解决带有不确定参数决策问题的一种有效方法，通过构建参数分布的不确定集合，将未来的各种不确定性在预先指定的置信水平下加以定量描述，使得决策结果更平稳，从而降低模型对输入参数扰动的敏感性 [2] 。

2. 单一金融资产收益率分布的GARCH-EVT模型

2.1. GARCH模型

GARCH模型由条件均值方程和条件方差方程组成。GARCH(p,q)模型的一般结构如下：

$\left\{\begin{array}{l}{r}_{t}={\mu }_{t}+{\epsilon }_{t}\\ {\epsilon }_{t}={\sigma }_{t}\cdot {z}_{t}\\ {\sigma }_{t}^{2}={\alpha }_{0}+{\sum }_{i=1}^{p}{\alpha }_{i}{\sigma }_{t-i}^{2}+{\sum }_{j=1}^{q}{\beta }_{j}{\epsilon }_{t-j}^{2}\\ {\alpha }_{0}>0,\text{\hspace{0.17em}}{\alpha }_{i}\ge 0,\text{\hspace{0.17em}}{\beta }_{j}\ge 0,\text{\hspace{0.17em}}{\sum }^{\text{​}}\text{ }{\alpha }_{i}+{\beta }_{j}<1\end{array}$ (1)

2.2. EVT模型

${F}_{u}\left(y\right)=Pr\left(X-uu\right)=\frac{F\left(u+y\right)-F\left(u\right)}{1-F\left(u\right)}=\frac{F\left(x\right)-F\left(u\right)}{1-F\left( u \right)}$

${G}_{\xi ,\beta }\left(y\right)=\left\{\begin{array}{l}1-{\left(1+\frac{\xi }{\beta }y\right)}^{-\frac{1}{\xi }},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\xi \ne 0\\ 1-\mathrm{exp}\left(-\frac{y}{\beta }\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\xi =0\end{array}$

2.3. GARCH-EVT模型

$\left({z}_{i,t-m+1},\cdots ,{z}_{i,t-1},{z}_{i,t}\right)=\left(\frac{{R}_{i,t-m+1}-{\stackrel{^}{\mu }}_{i,t-m+1}}{{\stackrel{^}{\sigma }}_{i,t-m+1}},\cdots ,\frac{{R}_{i,t-1}-{\stackrel{^}{\mu }}_{i,t-1}}{{\stackrel{^}{\sigma }}_{i,t-1}},\frac{{R}_{i,t}-{\stackrel{^}{\mu }}_{i,t}}{{\stackrel{^}{\sigma }}_{i,t}}\right)$

$F\left(z\right)=\left\{\begin{array}{l}\frac{{T}_{{u}^{L}}}{T}{\left\{1+{\xi }^{L}\frac{{u}^{L}-z}{{\beta }^{L}}\right\}}^{-\frac{1}{{\xi }^{L}}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z<{u}^{L}\\ \Phi \left(z\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}^{L}\le z\le {u}^{R}\\ 1-\frac{{T}_{{u}^{R}}}{T}{\left\{1+{\xi }^{R}\frac{z-{u}^{R}}{{\beta }^{R}}\right\}}^{-\frac{1}{{\xi }^{R}}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z>{u}^{R}\end{array}$ (2)

3. 多种金融资产收益率分布的GARCH-EVT-Copula模型

3.1. Copula函数

Copula函数可以分为椭圆族Copula和阿基米德族Copula。椭圆族Copula包括正态Copula函数以及t-Copula函数。椭圆Copula适合描述具有对称相依结构的数据，但是在金融领域，许多金融数据表现出非对称、非线性相关的特点。Genest和Mackay (1986) [10] 提出的阿基米德Copula函数可以有效的捕捉多元分布中的非对称非线性相依结构。阿基米德Copula函数包括Frank Copula、Gumbel Copula以及Clayton Copula。Frank Copula函数具有对称性，它无法捕捉到随机变量间的非对称相关；Gumbel Copula对变量在分布上尾部的变化十分敏感，能够快速捕捉到上尾相关的变化，但对在下尾部的变化不敏感；Clayton Copula对变量在分布下尾部的变化十分敏感，能够快速捕捉到下尾相关的变化。

3.2. GARCH-EVT-Copula模型

4. GARCH-EVT-Copula下的鲁棒投资组合模型

$x\in {R}^{n}$ 表示随机收益率向量， $w\in {R}^{n}$ 表示决策向量， $f\left(w,x\right)$ 表示相应的损失函数。假设 $x$ 服从密度函数为 $p\left(.\right)$ 的连续分布。

$\Psi \left(w,\alpha \right)\triangleq {{\int }^{\text{​}}}_{f\left(w,x\right)\le \alpha }p\left(x\right)\text{d}x$

$Va{R}_{\beta }\left(w\right)\triangleq \mathrm{min}\left\{\alpha \in R:\Psi \left(w,\alpha \right)\ge \beta \right\}$

$CVa{R}_{\beta }\left(w\right)\triangleq \frac{1}{1-\beta }{{\int }^{\text{​}}}_{f\left(w,x\right)\ge Va{R}_{\beta }\left(w\right)}f\left(w,x\right)p\left(x\right)\text{d}x$

Rockafellar和Uryasev (2003) [11] 证明了对于CVaR的计算可以由如下公式代替：

${F}_{\beta }\left(w,\alpha \right)\triangleq \alpha +\frac{1}{1-\beta }{{\int }^{\text{​}}}_{x\in {R}^{n}}{\left[f\left(w,x\right)-\alpha \right]}^{+}p\left(x\right)\text{d}x$ (3)

$CVa{R}_{\beta }\left(w\right)={\mathrm{min}}_{a\in R}{F}_{\beta }\left(w,\alpha \right)$

Kakouris和Rustem (2014) [4] 研究了用Coupla理论构建的多元分布下的VaR和CVaR，有如下结论：

$Va{R}_{\beta }\left(w\right)\triangleq \mathrm{min}\left\{\alpha \in R:c\left(u|\stackrel{˜}{f}\left(w,u\right)\le \alpha \right)\ge \beta \right\}$

$CVa{R}_{\beta }\left(w\right)\triangleq \frac{1}{1-\beta }{{\int }^{\text{​}}}_{\stackrel{˜}{f}\left(w,u\right)\ge Va{R}_{\beta }\left(w\right)}\stackrel{˜}{f}\left(w,u\right)c\left(u\right)\text{d}u$

${Y}_{\beta }\left(w,\alpha \right)=\alpha +\frac{1}{1-\beta }{{\int }^{\text{​}}}_{u\in {I}^{n}}{\left[\stackrel{˜}{f}\left(w,u\right)-\alpha \right]}^{+}c\left(u\right)\text{d}u$

$CVa{R}_{\beta }\left(w\right)={\mathrm{min}}_{a\in R}{Y}_{\beta }\left(w,\alpha \right)$

$H\left(x\right)\triangleq \left\{{\sum }_{i=1}^{l}{\lambda }_{i}{c}_{i}\left(F\left(x\right)\right):\lambda =\left({\lambda }_{1},\cdots ,{\lambda }_{l}\right)\in \Lambda ,{c}_{i}\in C,i=1,\cdots ,l\right\}$

$CVa{R}_{\beta }\left(w\right)={\mathrm{min}}_{a\in R}{Y}_{\beta }\left(w,\alpha \right)$

${Y}_{\beta }\left(w,\alpha \right)=\alpha +\frac{1}{1-\beta }{{\int }^{\text{​}}}_{u\in {I}^{n}}{\left[\stackrel{˜}{f}\left(w,u\right)-\alpha \right]}^{+}{\sum }_{i=1}^{l}{\lambda }_{i}{c}_{i}\left(u\right)\text{d}u={\sum }_{i=1}^{l}{\lambda }_{i}{Y}_{\beta }^{i}\left(w,\alpha \right)$

$WCVa{R}_{\beta }\left(w\right)\triangleq {\mathrm{sup}}_{c\left(.\right)\in C}CVa{R}_{\beta }\left( w \right)$

$WCVa{R}_{\beta }\left(w\right)={\mathrm{min}}_{a\in R}{\mathrm{max}}_{\lambda \in \Lambda }{Y}_{\beta }\left(w,\alpha \right)$

${\mathrm{min}}_{w\in W}WCVa{R}_{\beta }\left(w\right)={\mathrm{min}}_{w\in W}{\mathrm{min}}_{a\in R}{\mathrm{max}}_{\lambda \in \Lambda }{Y}_{\beta }\left(w,\alpha \right)$

${\mathrm{min}}_{\left(w,\alpha ,\theta \right)\in W×R×R}\left\{\theta :{\sum }_{i=1}^{l}{\lambda }_{i}{Y}_{\beta }^{i}\left(w,\alpha \right)\le \theta ,\lambda \in \Lambda \right\}$

Zhu和Fukushima (2009) [3] 证明了上述模型可以进一步简化为如下形式：

${\mathrm{min}}_{\left(w,\alpha ,\theta \right)\in W×R×R}\left\{\theta :{Y}_{\beta }^{i}\left(w,\alpha \right)\le \theta ,i=1,\cdots ,l\right\}$ (4)

${\mathrm{min}}_{w\in {R}^{n},v\in {R}^{l},\alpha \in R,\theta \in R}\theta$

$s.t.\text{\hspace{0.17em}}w\in W,v\in {R}^{m},\alpha \in R,\theta \in R$

$\alpha +\frac{1}{{S}^{i}\left(1-\beta \right)}{\left({1}^{i}\right)}^{\text{T}}{v}^{i}\le \theta ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\cdots ,l$

${v}_{k}^{i}\ge \stackrel{˜}{f}\left(w,{u}_{k}^{i}\right)-\alpha ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1,\cdots ,{S}^{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\cdots ,l$

${v}_{k}^{i}\ge 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1,\cdots ,{S}^{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\cdots ,l$

5. 实证研究

5.1. 数据

Figure 1. SSE 180’s energy and industrial industry index data

Figure 2. The log-return on the energy and industry index of SSE 180

5.2. 实证结果

5.2.1. 静态投资组合策略

Table 1. The descriptive statistics and JB test of the diurnal log-yield

$\left\{\begin{array}{l}{r}_{1,t}=-0.0313{r}_{1,t-1}-0.0311{r}_{1,t-2}-0.0048{r}_{1,t-3}+0.0336{r}_{1,t-4}+0.0714{ϵ}_{1,t-1}+{\epsilon }_{1,t}\\ {\epsilon }_{1,t}={\sigma }_{1,t}\cdot {z}_{1,t},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{z}_{1,t}~{F}_{1}\left(z\right)\\ {\sigma }_{1,t}^{2}=0.0575{\sigma }_{1,t-1}^{2}+0.9338{\epsilon }_{1,t-1}^{2}\end{array}$

${F}_{1}\left(z\right)=\left\{\begin{array}{l}0.0959\cdot {\left(1-0.1629\cdot \frac{-1.2076-z}{0.7225}\right)}^{\frac{1}{0.1629}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z<-1.2076\\ \Phi \left(z\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-1.2076\le z\le 1.2172\\ 0.9000\cdot {\left(1+0.0325\cdot \frac{z-1.2172}{0.6157}\right)}^{-\frac{1}{0.0325}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z>1.2172\end{array}$

$\left\{\begin{array}{l}{r}_{2,t}=-0.6649{r}_{2,t-1}+0.0018{r}_{2,t-2}+0.7050{ϵ}_{2,t-1}+{\epsilon }_{2,t}\\ {\epsilon }_{2,t}={\sigma }_{2,t}\cdot {z}_{2,t},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{z}_{2,t}~{F}_{2}\left(z\right)\\ {\sigma }_{2,t}^{2}=0.0558{\sigma }_{2,t-1}^{2}+0.9352{\epsilon }_{2,t-1}^{2}\end{array}$

${F}_{2}\left(z\right)=\left\{\begin{array}{l}0.0997\cdot {\left(1+0.0825\cdot \frac{-1.2190-z}{0.5814}\right)}^{-\frac{1}{0.0825}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z<-1.2190\\ \Phi \left(z\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-1.2190\le z\le 1.2074\\ 0.9003\cdot {\left(1+0.0184\cdot \frac{z-1.2074}{0.5169}\right)}^{-\frac{1}{0.0184}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z>1.2074\end{array}$

5.2.2. 动态投资组合策略

Table 2. Static investment portfolio strategy constructed by three models

Table 3. Comparison of investment portfolio built by three models

Figure 3. The cumulative log-return of three models

Figure 4. The cumulative log-return of the two robust models during the stock market crash

Figure 5. The log-return of SSE 180 energy and industrial index between 2016/1 and 2016/5

Figure 6. Cumulative log-return of two robust models between 2016/1 and 2016/5

6. 结论

[1] Black, F. and Litterman, R. (1992) Global Portfolio Optimization. Financial Analysts Journal, 48, 28-43.
https://doi.org/10.2469/faj.v48.n5.28

[2] 梁锡坤, 徐成贤, 郑冬, 等. 鲁棒投资组合选择优化问题的研究进展[J]. 运筹学学报, 2014, 18(2): 87-95.

[3] Zhu, S. and Fukushima, M. (2009) Worst-Case Conditional Value-at-Risk with Application to Robust Portfolio Management. Operations Research, 57, 1155-1168.
https://doi.org/10.1287/opre.1080.0684

[4] Kakouris, I. and Rustem, B. (2014) Robust Portfolio Optimization with Copulas. European Journal of Operational Research, 235, 28-37.
https://doi.org/10.1016/j.ejor.2013.12.022

[5] Bali, T.G. and Neftci, S.N. (2003) Disturbing Extremal Behavior of Spot Rate Dynamics. Journal of Empirical Finance, 10, 455-477.
https://doi.org/10.1016/S0927-5398(02)00070-1

[6] 刘志东, 徐淼. 基于GARCH和EVT的金融资产风险价值度量方法[J]. 统计与决策, 2007(18): 13-16.

[7] Jondeau, E. and Rockinger, M. (2003) Conditional Dependency of Financial Series: The Copula-GARCH Model. Social Science Electronic Publishing, rp69, 853.
https://doi.org/10.2139/ssrn.410740

[8] 李秀敏, 史道济. 金融市场组合风险的相关性研究[J]. 系统工程理论与实践, 2007, 27(2): 112-117.

[9] 张进滔. 基于GARCH-EVT方法和Copula函数的组合风险分析[D]: [硕士学位论文]. 成都: 四川大学, 2007.

[10] Genest, C. and Mackay, J. (1986) The Joy of Copulas: Bivariate Distributions with Uniform Marginals. American Statistician, 40, 280-283.

[11] Rockafellar, R.T. and Uryasev, S. (2002) Conditional Value-at-Risk for General Loss Distributions. Journal of Banking & Finance, 26, 1443-1471.
https://doi.org/10.1016/S0378-4266(02)00271-6

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