The use of copulas in risk management.

**Authors:**Stander, Yolanda Sophia**Date:**2012-08-15**Subjects:**Copulas (Mathematical statistics) , Dependence (Statistics) , Risk management**Type:**Mini-Dissertation**Identifier:**uj:9418 , http://hdl.handle.net/10210/5852**Description:**M.Sc. , In this dissertation we take a closer look at how copulas can be used to improve the risk measurement at a financial institution. The focus is on market risk in a trading environment. In practice risk numbers are calculated with very basic measures that are easy to explain to senior management and to traders. It is important that traders understand the risk measure as that helps them to understand the risk inherent in any deal and may assist them in deciding on the optimal hedge. The purpose of a hedge is to reduce the risk in a portfolio. As senior management is responsible for deciding on the optimal risk limits and risk appetite of the financial institution, it is important for them to understand what the risks are and how to measure these. The simplicity of the risk measures leads to certain inadequacies that can have very negative consequences for a financial institution. If the risk measure does not adequately capture the risk of a deal, the financial institution may suffer big losses when there are stress events in the market. Alternatively, when the risk measure overestimates the risk of a deal, too much economic capital is tied up in the deal. This inhibits the trader from adding more deals to a portfolio that may potentially lead to big profits. Economic capital is the capital that has to be held against positions to protect the financial institution if and when extreme market moves occur. In this dissertation the focus is on how copulas can be used to improve current risk measures. We focus on bivariate copulas. Bivariate copulas are easier to depict graphically than multivariate copulas with more than two dimensions. It is also easier to prove that the fitted bivariate copulae do adequately describe the underlying dependence structure between risk factors. Even though the focus is on the bivariate case, all methodologies can easily be extended to higher dimensions. In Chapter 1 copulas are defined and some basic copula properties are shown. We consider the definition of elliptical copulas and discuss some drawbacks to using them in a financial application. Some useful Archimedean copula properties are discussed and it is shown how to generate the copula function for n 2 dimensions. The various ways in which to estimate the parameters of a copula are also discussed as well as goodness-of-fit tests that are used to test whether the copula fits the underlying data adequately. Finally the chapter ends with an example that illustrates the theory. A back-test is done to establish whether the copula adequately describes the dependence structure over time. It is also shown how the fitted copula can be used to generate stress scenarios that are used as an alternative to historical scenarios when calculating a value-at-risk (VaR) number. In chapter 2 the properties of a dependence measure are discussed and it is argued that linear correlation does not conform to these desired properties. Rank correlation measures have some additional properties that make them more efficient than linear correlation measures in certain instances. We also consider their relationship to copulas. Finally it is shown how copulas can be used in practice to get another view on the dependence structure between risk factors. In risk measurement we are mainly concerned with extreme moves that market variables may show. In chapter 3 some of the techniques used in risk management are discussed as well as some of their shortcomings. The shortcomings are addressed by applying extreme value theory to calculate stress factors and using copulas to model the dependence structure between risk factors. The theory underlying bivariate extreme copulas is discussed and illustrated with a practical example.**Full Text:**

**Authors:**Stander, Yolanda Sophia**Date:**2012-08-15**Subjects:**Copulas (Mathematical statistics) , Dependence (Statistics) , Risk management**Type:**Mini-Dissertation**Identifier:**uj:9418 , http://hdl.handle.net/10210/5852**Description:**M.Sc. , In this dissertation we take a closer look at how copulas can be used to improve the risk measurement at a financial institution. The focus is on market risk in a trading environment. In practice risk numbers are calculated with very basic measures that are easy to explain to senior management and to traders. It is important that traders understand the risk measure as that helps them to understand the risk inherent in any deal and may assist them in deciding on the optimal hedge. The purpose of a hedge is to reduce the risk in a portfolio. As senior management is responsible for deciding on the optimal risk limits and risk appetite of the financial institution, it is important for them to understand what the risks are and how to measure these. The simplicity of the risk measures leads to certain inadequacies that can have very negative consequences for a financial institution. If the risk measure does not adequately capture the risk of a deal, the financial institution may suffer big losses when there are stress events in the market. Alternatively, when the risk measure overestimates the risk of a deal, too much economic capital is tied up in the deal. This inhibits the trader from adding more deals to a portfolio that may potentially lead to big profits. Economic capital is the capital that has to be held against positions to protect the financial institution if and when extreme market moves occur. In this dissertation the focus is on how copulas can be used to improve current risk measures. We focus on bivariate copulas. Bivariate copulas are easier to depict graphically than multivariate copulas with more than two dimensions. It is also easier to prove that the fitted bivariate copulae do adequately describe the underlying dependence structure between risk factors. Even though the focus is on the bivariate case, all methodologies can easily be extended to higher dimensions. In Chapter 1 copulas are defined and some basic copula properties are shown. We consider the definition of elliptical copulas and discuss some drawbacks to using them in a financial application. Some useful Archimedean copula properties are discussed and it is shown how to generate the copula function for n 2 dimensions. The various ways in which to estimate the parameters of a copula are also discussed as well as goodness-of-fit tests that are used to test whether the copula fits the underlying data adequately. Finally the chapter ends with an example that illustrates the theory. A back-test is done to establish whether the copula adequately describes the dependence structure over time. It is also shown how the fitted copula can be used to generate stress scenarios that are used as an alternative to historical scenarios when calculating a value-at-risk (VaR) number. In chapter 2 the properties of a dependence measure are discussed and it is argued that linear correlation does not conform to these desired properties. Rank correlation measures have some additional properties that make them more efficient than linear correlation measures in certain instances. We also consider their relationship to copulas. Finally it is shown how copulas can be used in practice to get another view on the dependence structure between risk factors. In risk measurement we are mainly concerned with extreme moves that market variables may show. In chapter 3 some of the techniques used in risk management are discussed as well as some of their shortcomings. The shortcomings are addressed by applying extreme value theory to calculate stress factors and using copulas to model the dependence structure between risk factors. The theory underlying bivariate extreme copulas is discussed and illustrated with a practical example.**Full Text:**

Dependence analysis in BRICS stock markets : a vine copula approach

**Authors:**Tang, Liang**Date:**2019**Subjects:**Stock exchanges - BRIC countries , Stocks - Prices - BRIC countries , Dependence (Statistics) , Copulas (Mathematical statistics)**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/402948 , uj:33743**Description:**Abstract : This study makes use of three types of vine copulas, c-vine, d-vine and r-vine copulas, to investigate the dependence structure in the BRICS stock markets using daily stock market price data spanning from 28-12-2000 to 10-08-2018. To account for the dynamic effects in dependence measures, the study divides the sample period into three sub-samples: the pre-crisis period (from 28-12- 2000 to 31-01-2007), the crisis period (from 01-02-2007 to 29-12-2011), and the post-crisis period (from 04-01-2012 to 10-08-2018). The price data is first converted to return series and filtered using different ARIMA-GARCH models in order to remove the autocorrelation and heteroscedasticity effects. During this process, it was found that most of the return series exhibited leverage effects, an indication that bad news in the stock markets leads to larger spikes in volatility than good news does. To understand the implication of this effect on the dependence structure of stock markets in the BRICS countries, the c-vine, d-vine and r-vine copulas are used. The use of vine copulas has some significant advantages over traditional copulas as they model the dependence in the BRICS using pairwise copula constructions. The results show that the three types of vine copula models suggest that Student’s t and the SBB7 copulas best describe the dependence structure in the BRICS markets. Unlike other studies, our findings show the existence of a very strong dependence between South Africa and Russia, South Africa and India, and South Africa and Brazil during the pre-crisis, the crisis and the post-crisis periods, suggesting a financial integration between these three countries. Furthermore, we find strong dependence between China and the rest of BRICS markets only during a financial crisis. The study identifies two types of dependence in the BRICS stock markets: the first is among small economies (South Africa, Brazil and Russia) and the second one among large economies (China and India). Small economies tend to co-move during bull and bear markets while large economies co-move with the rest only during bear market periods. , M.Com. (Financial Economics)**Full Text:**

**Authors:**Tang, Liang**Date:**2019**Subjects:**Stock exchanges - BRIC countries , Stocks - Prices - BRIC countries , Dependence (Statistics) , Copulas (Mathematical statistics)**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/402948 , uj:33743**Description:**Abstract : This study makes use of three types of vine copulas, c-vine, d-vine and r-vine copulas, to investigate the dependence structure in the BRICS stock markets using daily stock market price data spanning from 28-12-2000 to 10-08-2018. To account for the dynamic effects in dependence measures, the study divides the sample period into three sub-samples: the pre-crisis period (from 28-12- 2000 to 31-01-2007), the crisis period (from 01-02-2007 to 29-12-2011), and the post-crisis period (from 04-01-2012 to 10-08-2018). The price data is first converted to return series and filtered using different ARIMA-GARCH models in order to remove the autocorrelation and heteroscedasticity effects. During this process, it was found that most of the return series exhibited leverage effects, an indication that bad news in the stock markets leads to larger spikes in volatility than good news does. To understand the implication of this effect on the dependence structure of stock markets in the BRICS countries, the c-vine, d-vine and r-vine copulas are used. The use of vine copulas has some significant advantages over traditional copulas as they model the dependence in the BRICS using pairwise copula constructions. The results show that the three types of vine copula models suggest that Student’s t and the SBB7 copulas best describe the dependence structure in the BRICS markets. Unlike other studies, our findings show the existence of a very strong dependence between South Africa and Russia, South Africa and India, and South Africa and Brazil during the pre-crisis, the crisis and the post-crisis periods, suggesting a financial integration between these three countries. Furthermore, we find strong dependence between China and the rest of BRICS markets only during a financial crisis. The study identifies two types of dependence in the BRICS stock markets: the first is among small economies (South Africa, Brazil and Russia) and the second one among large economies (China and India). Small economies tend to co-move during bull and bear markets while large economies co-move with the rest only during bear market periods. , M.Com. (Financial Economics)**Full Text:**

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