An application of vine copula to the portfolio optimisation problem

**Authors:**Sithole, Pansy Rumbidzai**Date:**2015**Subjects:**Copulas (Mathematical statistics)**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/54894 , uj:16255**Description:**Abstract: It is often very difficult to accurately measure dependence structure in multivariate distributions that exhibit asymmetry and heavy tail dependence. Hence, the use of copula models which have become popular nowadays because they take into account the skewness and tail dependence of asset returns. However, the use of copula is challenging in higher dimensions, because of inflexible structures. Thus, this study focuses on the decomposition of a multivariate distribution model into pairwise copula. Although vine copulas have many dimensions, this study focuses on the C- and D-vine models. Moreover, the research examines the application of the C- and D-vine models in portfolio optimization. The examination considers 6 JSE stock indices, namely, Beverage Index, Construction and Materials Index, Financial and Industrial 30, Healthcare Index, Mining Index, and Telecommunication Index – with daily data spanning from 05 January 1998 to 15 October 2014. The results of the Sharpe ratio indicate that the C- and D-vine copulas are better models for measuring the dependence structure in portfolio optimization. This is because these two models are able to decompose a multivariate probability density into bivariate copulas, thereby allowing the different structural behaviours of the pairs of variables to be modeled accurately. Similar results for the application of vine copula in portfolio optimisation have been shown in the literature review. This is evidence that the C- and D-vine copula models can be implemented in South Africa to measure accurately the dependence between returns with high excess skewness and kurtosis on a daily basis. , M.Com.**Full Text:**

**Authors:**Sithole, Pansy Rumbidzai**Date:**2015**Subjects:**Copulas (Mathematical statistics)**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/54894 , uj:16255**Description:**Abstract: It is often very difficult to accurately measure dependence structure in multivariate distributions that exhibit asymmetry and heavy tail dependence. Hence, the use of copula models which have become popular nowadays because they take into account the skewness and tail dependence of asset returns. However, the use of copula is challenging in higher dimensions, because of inflexible structures. Thus, this study focuses on the decomposition of a multivariate distribution model into pairwise copula. Although vine copulas have many dimensions, this study focuses on the C- and D-vine models. Moreover, the research examines the application of the C- and D-vine models in portfolio optimization. The examination considers 6 JSE stock indices, namely, Beverage Index, Construction and Materials Index, Financial and Industrial 30, Healthcare Index, Mining Index, and Telecommunication Index – with daily data spanning from 05 January 1998 to 15 October 2014. The results of the Sharpe ratio indicate that the C- and D-vine copulas are better models for measuring the dependence structure in portfolio optimization. This is because these two models are able to decompose a multivariate probability density into bivariate copulas, thereby allowing the different structural behaviours of the pairs of variables to be modeled accurately. Similar results for the application of vine copula in portfolio optimisation have been shown in the literature review. This is evidence that the C- and D-vine copula models can be implemented in South Africa to measure accurately the dependence between returns with high excess skewness and kurtosis on a daily basis. , M.Com.**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:**

Estimating portfolio value at risk by a conditional copula approach in BRICS countries

- Mukalenge, Tshikenda Leopold

**Authors:**Mukalenge, Tshikenda Leopold**Date:**2018**Subjects:**Portfolio management - BRIC countries , Financial risk management - BRIC countries - Statistical methods , Copulas (Mathematical statistics)**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/403400 , uj:33802**Description:**Abstract : This thesis used daily log returns of indices of BRICS countries from the period of March 11th 2013 to May 16th 2017. Its main focus was to estimate the value at risk (VaR) of a portfolio of the BRICS financial markets using a conditional copula approach. A useful starting point was to apply the model of AR (1)-GARCH (1,1) with t-distribution and AR (1)-GARCH (1,1), using returns of the normal errors for the marginal distribution models in the copula framework. Two copulas, the normal and the symmetric Joe Clayton (SJC) copulas, were estimated as both constant and time-varying. The log likelihood of the time-varying copula was significantly more suitable than the constant copula. The comparison of the performance of the copula models to the benchmark AR (1)-GARCH (1,1) was done using the Christoffersen test. The 99% VaR appeared fairly accurate, suggesting that the VaR models were dependable. The standard level of comparison AR (1)-GARCH (1,1) did not perform well compared to the SJC copula; i.e. the time-varying SJC copula performed better than the benchmark model. The time-varying SJC copula model used to estimate the portfolio VaR also showed a minimum number of exceptions in the back-test. This copula thus meets regulatory capital requirement for investors as stipulated in Basel II. , M.Com. (Financial Economics)**Full Text:**

**Authors:**Mukalenge, Tshikenda Leopold**Date:**2018**Subjects:**Portfolio management - BRIC countries , Financial risk management - BRIC countries - Statistical methods , Copulas (Mathematical statistics)**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/403400 , uj:33802**Description:**Abstract : This thesis used daily log returns of indices of BRICS countries from the period of March 11th 2013 to May 16th 2017. Its main focus was to estimate the value at risk (VaR) of a portfolio of the BRICS financial markets using a conditional copula approach. A useful starting point was to apply the model of AR (1)-GARCH (1,1) with t-distribution and AR (1)-GARCH (1,1), using returns of the normal errors for the marginal distribution models in the copula framework. Two copulas, the normal and the symmetric Joe Clayton (SJC) copulas, were estimated as both constant and time-varying. The log likelihood of the time-varying copula was significantly more suitable than the constant copula. The comparison of the performance of the copula models to the benchmark AR (1)-GARCH (1,1) was done using the Christoffersen test. The 99% VaR appeared fairly accurate, suggesting that the VaR models were dependable. The standard level of comparison AR (1)-GARCH (1,1) did not perform well compared to the SJC copula; i.e. the time-varying SJC copula performed better than the benchmark model. The time-varying SJC copula model used to estimate the portfolio VaR also showed a minimum number of exceptions in the back-test. This copula thus meets regulatory capital requirement for investors as stipulated in Basel II. , M.Com. (Financial Economics)**Full Text:**

Linear predictor of discounted aggregated cash flows with dependent inter-occurrence time

**Authors:**Shipalana, Peace Victory**Date:**2019**Subjects:**Finance - Mathematical models , Accounting - Mathematical models , Copulas (Mathematical statistics) , Portfolio management**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/403013 , uj:33751**Description:**Abstract : In this minor dissertation we derive the first two moments and a linear predictor of the compound discounted renewal aggregate cash flows when taking into account dependence within the inter-occurrence times. To illustrate our results, we use specific mixtures of exponential distributions to define the Archimedean copula, the dependence structure between the cash flow inter-occurrence times. The Ho-Lee interest rate model is used to show that the formulas derived can be calculated. , M.Com. (Financial Economics)**Full Text:**

**Authors:**Shipalana, Peace Victory**Date:**2019**Subjects:**Finance - Mathematical models , Accounting - Mathematical models , Copulas (Mathematical statistics) , Portfolio management**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/403013 , uj:33751**Description:**Abstract : In this minor dissertation we derive the first two moments and a linear predictor of the compound discounted renewal aggregate cash flows when taking into account dependence within the inter-occurrence times. To illustrate our results, we use specific mixtures of exponential distributions to define the Archimedean copula, the dependence structure between the cash flow inter-occurrence times. The Ho-Lee interest rate model is used to show that the formulas derived can be calculated. , M.Com. (Financial Economics)**Full Text:**

Modelling aggregate risk of the South African banking industry in the context of the Basil Pillar II framework

**Authors:**Khoza, Dingaan Jack**Date:**2017**Subjects:**Banks and banking - South Africa , Banks and banking - Risk management - South Africa , Bank capital - South Africa , Financial statements - South Africa , Copulas (Mathematical statistics)**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/245834 , uj:25470**Description:**M.Com. , Abstract: This study uses the aggregate balance sheet and income statement of South African banks to implement a risk aggregation model that aggregates credit, market and operational risks with the aim of generating total risk estimates using both Value at Risk (VaR) and Expected Shortfall (ES) as risk measures. The results are thereafter used to determine the supplemental Pillar II economic capital required in order to maintain the capital adequacy of the South African banking industry. We first model the return distributions due to credit and market risk using a multivariate risk factors sensitivity model, with the macroeconomic risk factors’ dynamics modelled through an asymmetric GARCH (generalize Autoregressive Conditional Heteroskedasticity) model designed by Baba, Engle, Kraft and Krona (1990) (i.e. BEKK). Operational risk losses are assumed to follow a lognormal distribution. The Gaussian copula and t-copulas are then used to aggregate the three loss distributions (i.e. credit, market and operational risk distributions). The total risk given by copulas is compared to the total risk calculated through the less complex simple additive and variance-covariance methods. Our results suggest that the South African banking sector’s Pillar I regulatory capital as at end of December 2015 should be supplemented by an amount of approximately 52 billion ZAR when using as a benchmark the Gaussian copula risk aggregation model measured through the ES metric at 99.9% confidence level. These results suggest that the Pillar 2A capital requirement imposed by the SARB should double from the current maximum of 2% to 4%.**Full Text:**

**Authors:**Khoza, Dingaan Jack**Date:**2017**Subjects:**Banks and banking - South Africa , Banks and banking - Risk management - South Africa , Bank capital - South Africa , Financial statements - South Africa , Copulas (Mathematical statistics)**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/245834 , uj:25470**Description:**M.Com. , Abstract: This study uses the aggregate balance sheet and income statement of South African banks to implement a risk aggregation model that aggregates credit, market and operational risks with the aim of generating total risk estimates using both Value at Risk (VaR) and Expected Shortfall (ES) as risk measures. The results are thereafter used to determine the supplemental Pillar II economic capital required in order to maintain the capital adequacy of the South African banking industry. We first model the return distributions due to credit and market risk using a multivariate risk factors sensitivity model, with the macroeconomic risk factors’ dynamics modelled through an asymmetric GARCH (generalize Autoregressive Conditional Heteroskedasticity) model designed by Baba, Engle, Kraft and Krona (1990) (i.e. BEKK). Operational risk losses are assumed to follow a lognormal distribution. The Gaussian copula and t-copulas are then used to aggregate the three loss distributions (i.e. credit, market and operational risk distributions). The total risk given by copulas is compared to the total risk calculated through the less complex simple additive and variance-covariance methods. Our results suggest that the South African banking sector’s Pillar I regulatory capital as at end of December 2015 should be supplemented by an amount of approximately 52 billion ZAR when using as a benchmark the Gaussian copula risk aggregation model measured through the ES metric at 99.9% confidence level. These results suggest that the Pillar 2A capital requirement imposed by the SARB should double from the current maximum of 2% to 4%.**Full Text:**

Moments of the discounted renewal cash flows : a copula approach

**Authors:**Dziwa, Simbarashe K.**Date:**2017**Subjects:**Copulas (Mathematical statistics) , Variables (Mathematics) , Financial risk management , Collateralized debt obligations**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/283402 , uj:30561**Description:**M.Com. (Financial Economics) , Abstract: Please refer to full text to view abstract.**Full Text:**

**Authors:**Dziwa, Simbarashe K.**Date:**2017**Subjects:**Copulas (Mathematical statistics) , Variables (Mathematics) , Financial risk management , Collateralized debt obligations**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/283402 , uj:30561**Description:**M.Com. (Financial Economics) , Abstract: Please refer to full text to view abstract.**Full Text:**

Multi-period portfolio optimization : a differential evolution copula-based approach

**Authors:**Mba, Jules Clement**Date:**2019**Subjects:**Copulas (Mathematical statistics) , Data encryption (Computer science) , Econometrics , Algorithms , Finance - Mathematical models**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/295977 , uj:32240**Description:**Abstract: Please refer to full text to view abstract. , M.Com. (Financial Economics)**Full Text:**

**Authors:**Mba, Jules Clement**Date:**2019**Subjects:**Copulas (Mathematical statistics) , Data encryption (Computer science) , Econometrics , Algorithms , Finance - Mathematical models**Language:**English**Type:**Masters (Thesis)**Identifier:**http://hdl.handle.net/10210/295977 , uj:32240**Description:**Abstract: Please refer to full text to view abstract. , M.Com. (Financial Economics)**Full Text:**

Multivariate copulas in financial market risk with particular focus on trading strategies and asset allocation

**Authors:**Stander, Yolanda Sophia**Date:**2012-11-05**Subjects:**Copulas (Mathematical statistics) , Variables (Mathematics) , Asset allocation , Financial risk**Type:**Thesis**Identifier:**uj:7351 , http://hdl.handle.net/10210/8099**Description:**D.Comm. , Copulas provide a useful way to model different types of dependence structures explicitly. Instead of having one correlation number that encapsulates everything known about the dependence between two variables, copulas capture information on the level of dependence as well as whether the two variables exhibit other types of dependence, for example tail dependence. Tail dependence refers to the instance where the variables show higher dependence between their extreme values. A copula is defined as a multivariate distribution function with uniform marginals. A useful class of copulas is known as Archimedean copulas that are constructed from generator functions with very specific properties. The main aim of this thesis is to construct multivariate Archimedean copulas by nesting different bivariate Archimedean copulas using the vine construction approach. A characteristic of the vine construction is that not all combinations of generator functions lead to valid multivariate copulas. Established research is limited in that it presents constraints that lead to valid multivariate copulas that can be used to model positive dependence only. The research in this thesis extends the theory by deriving the necessary constraints to model negative dependence as well. Specifically, it ensures that the multivariate copulas that are constructed from bivariate copulas that capture negative dependence, will be able to capture negative dependence as well. Constraints are successfully derived for trivariate copulas. It is, however, shown that the constraints cannot easily be extended to higher-order copulas. The rules on the types of dependence structures that can be nested are also established. A number of practical applications in the financial markets where copula theory can be utilized to enhance the more established methodologies, are considered. The first application considers trading strategies based on statistical arbitrage where the information in the bivariate copula structure is utilised to identify trading opportunities in the equity market. It is shown that trading costs adversely affect the profits generated. The second application considers the impact of wrong-way risk on counterparty credit exposure. A trivariate copula is used to model the wrong-way risk. The aim of the analysis is to show how the theory developed in this thesis should be applied where negative correlation is present in a trivariate copula structure. Approaches are considered where conditional and unconditional risk driver scenarios are derived by means of the trivariate copula structure. It is argued that by not allowing for wrong-way risk, a financial institution’s credit pricing and regulatory capital calculations may be adversely affected. The final application compares the philosophy behind cointegration and copula asset allocation techniques to test which approach produces the most profitable index-tracking portfolios over time. The copula asset allocation approach performs well over time; however, it is very computationally intensive.**Full Text:**

**Authors:**Stander, Yolanda Sophia**Date:**2012-11-05**Subjects:**Copulas (Mathematical statistics) , Variables (Mathematics) , Asset allocation , Financial risk**Type:**Thesis**Identifier:**uj:7351 , http://hdl.handle.net/10210/8099**Description:**D.Comm. , Copulas provide a useful way to model different types of dependence structures explicitly. Instead of having one correlation number that encapsulates everything known about the dependence between two variables, copulas capture information on the level of dependence as well as whether the two variables exhibit other types of dependence, for example tail dependence. Tail dependence refers to the instance where the variables show higher dependence between their extreme values. A copula is defined as a multivariate distribution function with uniform marginals. A useful class of copulas is known as Archimedean copulas that are constructed from generator functions with very specific properties. The main aim of this thesis is to construct multivariate Archimedean copulas by nesting different bivariate Archimedean copulas using the vine construction approach. A characteristic of the vine construction is that not all combinations of generator functions lead to valid multivariate copulas. Established research is limited in that it presents constraints that lead to valid multivariate copulas that can be used to model positive dependence only. The research in this thesis extends the theory by deriving the necessary constraints to model negative dependence as well. Specifically, it ensures that the multivariate copulas that are constructed from bivariate copulas that capture negative dependence, will be able to capture negative dependence as well. Constraints are successfully derived for trivariate copulas. It is, however, shown that the constraints cannot easily be extended to higher-order copulas. The rules on the types of dependence structures that can be nested are also established. A number of practical applications in the financial markets where copula theory can be utilized to enhance the more established methodologies, are considered. The first application considers trading strategies based on statistical arbitrage where the information in the bivariate copula structure is utilised to identify trading opportunities in the equity market. It is shown that trading costs adversely affect the profits generated. The second application considers the impact of wrong-way risk on counterparty credit exposure. A trivariate copula is used to model the wrong-way risk. The aim of the analysis is to show how the theory developed in this thesis should be applied where negative correlation is present in a trivariate copula structure. Approaches are considered where conditional and unconditional risk driver scenarios are derived by means of the trivariate copula structure. It is argued that by not allowing for wrong-way risk, a financial institution’s credit pricing and regulatory capital calculations may be adversely affected. The final application compares the philosophy behind cointegration and copula asset allocation techniques to test which approach produces the most profitable index-tracking portfolios over time. The copula asset allocation approach performs well over time; however, it is very computationally intensive.**Full Text:**

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:**

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