Measuring Portfolio Risks With Copula Theory

Financial risks usually happen with financial activities. There will be potential great risks when people are enjoying the benefits from the financial innovation and the financial liberalization. Thus people never forgot to pursuit a more reasonable and scientific measure method. From Markowitz's research on the trade-off relationship between the risks and returns of investment, to Sharp's further development of capital asset pricing model (CAPM) based on Markowitz ,and then Ross take the arbitrage pricing theory (APT), these initiative theoretical results have great influence on the research of the relationship between the risks and returns of investment. In 1994, J.P. Morgan first proposed value at risk (VaR) which have adopted by a number of financial institutions and it has been the most widely used method in the field of risk measure.The issue of the subprime mortgage of U.S. marked the formal outbreak of the financial crisis recently. This crisis brought lots of losses to American economy, and spread to the other economies rapidly, brought a heavy blow to the global economy. The strengthening of international contacts and the development of financial innovation cause the volatility of financial market increased, and the stability of the financial system decreased. Face the volatility of market, investor usually choose divert the investments to avoid the risks. The risk of portfolio is influenced by the factors of their own and also by factors of interactions each other. To take a reasonable measure of the risk, we should firstly establish the suitable multivariate function model. The classical multivariate function require the marginal distribution function must be the same type with the multivariate function. Moreover, the type of marginal distribution function must be the same. Faced with the more complex market situation, the marginal distribution functions of financial assets are usually not the same. Thus, the classical multivariate function will be not suit for giving a reasonable explanation to the dependence of the variables.The introduce of Copula in financial solves these problems successfully. Copula theory as a kind of nonlinear and asymmetric statistical theory be widely used in analysis of multivariate financial time series and risk management. Firstly , Copula can be used to construct multivariate flexibly, breaking the steep demand of the classical multivariable function. Secondly, the dependence of the Copula function which break the classical metric of linear dependence make a broader range of application of Copula. In addition, the Copula function can model the univariate marginals and the joint distribution separately. Therefore, the use of Copula can make it is easier to construct a multivariate function model. As the Copula can capture the nonlinear and asymmetric dependence of variables, the multivariate function models are closer to the real scenarios.This paper is divided into four chapters, as follow:The first chapter introduce the conditional heteroskedasticity models and Copula theory. SectionⅠ,we give the significance and the backgrounds of the paper. SectionⅡis the reviews of past literature about the conditional heteroskedasticity model, including ARCH and GARCH models and their development models. SectionⅢ,we introduce the Copula functions. Giving the development of Copula both abroad and domestic.The second chapter of this paper is the first part of theories. In this chapter, we introduce the method to model the marginal functions. Previous studies show that the conditional heteroskedasticity model perform well in modeling the financial indices. However,in the presence of asymmetry ,leptokurtosis, and fat tail characteristics of financial indicators, the classical GARCH model cannot fully simulate the complex market scenarios. Along the way to solve a complex problems, from simple to complex, we firstly introduce GJR-GARCH model which can capture the asymmetry characteristic of the market. And then, for the fat tail, we introduce EVT which simulate well in the tails of a distribution.The third chapter is the integrated part of the model which introduce the Copula theory. Section I ,we introduced the basic concept of Copula theory and the measure of dependence. Then ,introduced some commonly used Copulas. After that ,we introduced the simulation algorithm and maximum likelihood estimation of parameter estimation of the Copula function. In the sectionⅡ,we made a brief introduction about the concept of VaR, and then we give the measurement method of VaR based on Monte Carlo simulation.The fourth chapter, we choose six CSI 300 industry indices for the empirical study. According to the model that introduced before, established the model of portfolio. Find the VaR of the portfolio through the Monte Carlo simulation.We use GJR-GARCH and extreme value theory to model the marginals of the indices which fully taking into account the existence of asymmetry and fat tails in China's stock market. When the marginal distributions are relatively complicated, the application of Copula function to simulate their related structures, overcoming of the limitations of traditional linear multivariate distribution function.Empirical evidences show that the Copula theory is a useful attempt in the application of risk measurement. Copula function is actually a model of dependence structure. It is a powerful tool in financial. However, as a new theory in financial, there are more works to do both in theory and application.