﻿ CVaR度量在极值理论中的应用 Application of CVaR Metric in Extreme Value Theory

Pure Mathematics
Vol.06 No.02(2016), Article ID:17141,8 pages
10.12677/PM.2016.62014

Application of CVaR Metric in Extreme Value Theory

Jing Yao1, Yongming Li2

1Department of Mathematics, Guangxi Normal University, Nanning Guangxi

2Department of Mathematics, Shangrao Normal College, Shangrao Jiangxi

Received: Feb. 27th, 2016; accepted: Mar. 9th, 2016; published: Mar. 16th, 2016

ABSTRACT

Since the last half a century, with the globalization and diversification of economy, the financial risk measurement has gradually been concerned by the financial and economic scholars. After the 1990s, the new risk management tool, VaR (value at risk) measurement method has been developed gradually, which can measure risk value scientifically, accurately and comprehensively, and it is welcomed in the international financial community, but in extreme event, the accuracy of VaR is less than that of CVaR (conditional value at risk). This paper is intended to study the application of CVaR measure in extreme value theory.

Keywords:Extreme Value Theory, VaR, CVaR

CVaR度量在极值理论中的应用

1广西师范学院数学科学学院，广西 南宁

2上饶师范学院数学系，江西 上饶

1. 引言

2. 极值理论

θ为可测集，定义映射：。当同时满足下面的条件时，称为致性风险测度：

1) 单调性：

2) 次可加性：X，Y；

3) 正齐次性：

4) 传递性：

VaR不满足一致性测度公理的性质，CVaR和ES(Expected Shortfall)恰好能弥补VaR的不足。Acerbi (2002) [8] 给出当收益X的分布函数F(X)连续时，CVaR和ES等价。

CVaR(损失超出VaR条件期望)值为：

(1)

2.1. 极值分布

(2)

(3)

(4)

2.2. 极值CVaR的计算

(5)

A1：找到合适的阈值u。

A2：估计广义Pareto分布的参数采用极大似然估计。

A3：估计

A4：估计尾部及分位数。

(6)

3. 实证分析

Table 1. Basic statistics

Figure 1. Hill diagram

Figure 2. QQ diagram

Figure 3. Empirical distribution function

Figure 4. Simple average remaining figure

Figure 5. Estimate extreme value index

Figure 6. Parameters estimated GDP figure

Figure 7. GDP higher median estimated figure

Table 2. Risk measurement

4. 总结

Application of CVaR Metric in Extreme Value Theory[J]. 理论数学, 2016, 06(02): 95-102. http://dx.doi.org/10.12677/PM.2016.62014

1. 1. Rockafellar, T. and Uryasev, S. (2000) Optimization of Conditional Value-at-Risk. Jouranl of Risk, 2, 21-41.

2. 2. Andersson, F., Mausser, H., Rosen, D. and Uryasev, S. (2001) Credit Risk Optimization with Conditional Value-at- Risk Criterion. Mathematical Programming, 89, 273-291. http://dx.doi.org/10.1007/PL00011399

3. 3. 陈剑利, 李胜. CVaR风险度量模型在投资组合中的运用[J]. 运筹与管理, 2004(1): 95-99.

4. 4. 曲圣宁, 田新. 投资组合风险管理中的VaR模型的缺陷以及CVaR模型[J]. 统计与决策, 2005(10): 18-20.

5. 5. 霍玉琳, 何春雄. 研究了GARCH模型下的极值一致风险度量[J]. 金融经济, 2006(2): 152-154.

6. 6. 余星, 孙红果, 陈国华. 基于CVaR的融入期权的投资组合模型[J]. 数学的实践与认识, 2014(1): 11-14.

7. 7. 黄鹂, 魏岩. 基于CVaR模型投资组合保险的绩效实证[J]. 金融理论与实践, 2015(4): 98-103.

8. 8. Acerbi, C. and Tasche, D. (2002) On the Cohe-rence of Expected Shortfall. Journal of Banking and Finance, 26, 1487- 1503. http://dx.doi.org/10.1016/S0378-4266(02)00283-2