On the distortion risk measure using copulas

Ly, Sal; Pham, Uyen Hoang; Briš, Radim

doi:10.1201/b21348-51

Cited by 4 publications

(7 citation statements)

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“…We first discuss some work on copula methods that is related to the problem studied in this paper. Readers may refer to Ly et al (2016) for more information. For a weighted sum of two dependent random variables with special emphasis on the applications in estimating distortion risk measures and diversification, we assume that a portfolio Y that is a linear combination of two assets X 1 and X 2 with respect to their weights w 1 and w 2 is expressed as follows: Y = w 1 X 1 + w 2 X 2 with w 1 + w 2 = 1.…”

Section: Background Theorymentioning

confidence: 99%

“…Let F X 1 , F X 2 , and F Y denote the cumulative distribution functions (CDFs) of X 1 , X 2 , and Y, respectively. Suppose that investors are interested in estimating risks of the portfolio Y under distortion risk measure (see Ly et al (2016)), given by…”

Section: Background Theorymentioning

confidence: 99%

“…However, thus far, most studies only focus on independence structure with some common distributions of the functions of random variables (see, for instance Dettmann and Georgiou (2009) ;Garg et al (2016); Thompson (1966, 1970); Yang and Wang (2013)). There are few studies on determining distributions for statistical models involving dependence structures of some common distributions for several functions of random variables (see, for example Ly et al (2016Ly et al ( , 2019; Joe (1997)).…”

Section: Introductionmentioning

confidence: 99%

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Determining Distribution for the Product of Random Variables by Using Copulas

Pho

et al. 2019

Risks

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Determining distributions of the functions of random variables is one of the most important problems in statistics and applied mathematics because distributions of functions have wide range of applications in numerous areas in economics, finance, risk management, science, and others. However, most studies only focus on the distribution of independent variables or focus on some common distributions such as multivariate normal joint distributions for the functions of dependent random variables. To bridge the gap in the literature, in this paper, we first derive the general formulas to determine both density and distribution of the product for two or more random variables via copulas to capture the dependence structures among the variables. We then propose an approach combining Monte Carlo algorithm, graphical approach, and numerical analysis to efficiently estimate both density and distribution. We illustrate our approach by examining the shapes and behaviors of both density and distribution of the product for two log-normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas, and estimate some common measures including Kendall’s coefficient, mean, median, standard deviation, skewness, and kurtosis for the distributions. We found that different types of copulas affect the behavior of distributions differently. In addition, we also discuss the behaviors via all copulas above with the same Kendall’s coefficient. Our results are the foundation of any further study that relies on the density and cumulative probability functions of product for two or more random variables. Thus, the theory developed in this paper is useful for academics, practitioners, and policy makers.

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Section: Background Theorymentioning

confidence: 99%

Section: Background Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Determining Distribution for the Product of Random Variables by Using Copulas

Pho

et al. 2019

Risks

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“…where g is a distortion function and F Y 1 (y) = 1 − F Y 1 (y) is a survival function of Y 1 . Readers may refer to Ly et al (2016) for more detailed information. In the credit model, the total loss is defined as the aggregation of the product of risk factors.…”

Section: Background Theorymentioning

confidence: 99%

“…Determining distributions of the functions of random variables is a very crucial task and this problem has been attracted a number of researchers because there are numerous applications in Risk Management, Finance, Economics, Science, and, many other areas, see, for example, (Donahue 1964;Ly et al 2016;Nadarajah and Espejo 2006;Springer 1979). Basically, the distributions of an algebraic combination of random variables including the sum, product, and quotient are focused on some common distributions along with the assumptions of independence or correlated through Pearson's coefficient or dependence via multivariate normal joint distributions (Arnold and Brockett 1992;Bithas et al 2007;Cedilnik et al 2004;Hinkley 1969;Macalos and Arcede 2015;Marsaglia 1965;Matović et al 2013;Mekićet al 2012;Nadarajah and Espejo 2006;Kotz 2006a, 2006b;Pham-Gia et al 2006;Pham-Gia 2000;Rathie et al 2016;Sakamoto 1943).…”

Section: Introductionmentioning

confidence: 99%

Determining Distribution for the Quotients of Dependent and Independent Random Variables by Using Copulas

Pho

et al. 2019

JRFM

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Determining distributions of the functions of random variables is a very important problem with a wide range of applications in Risk Management, Finance, Economics, Science, and many other areas. This paper develops the theory on both density and distribution functions for the quotient Y = X 1 X 2 and the ratio of one variable over the sum of two variables Z = X 1 X 1 + X 2 of two dependent or independent random variables X 1 and X 2 by using copulas to capture the structures between X 1 and X 2 . Thereafter, we extend the theory by establishing the density and distribution functions for the quotients Y = X 1 X 2 and Z = X 1 X 1 + X 2 of two dependent normal random variables X 1 and X 2 in the case of Gaussian copulas. We then develop the theory on the median for the ratios of both Y and Z on two normal random variables X 1 and X 2 . Furthermore, we extend the result of median for Z to a larger family of symmetric distributions and symmetric copulas of X 1 and X 2 . Our results are the foundation of any further study that relies on the density and cumulative probability functions of ratios for two dependent or independent random variables. Since the densities and distributions of the ratios of both Y and Z are in terms of integrals and are very complicated, their exact forms cannot be obtained. To circumvent the difficulty, this paper introduces the Monte Carlo algorithm, numerical analysis, and graphical approach to efficiently compute the complicated integrals and study the behaviors of density and distribution. We illustrate our proposed approaches by using a simulation study with ratios of normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas. We find that copulas make big impacts from different Copulas on behavior of distributions, especially on median, spread, scale and skewness effects. In addition, we also discuss the behaviors via all copulas above with the same Kendall’s coefficient. The approaches developed in this paper are flexible and have a wide range of applications for both symmetric and non-symmetric distributions and also for both skewed and non-skewed copulas with absolutely continuous random variables that could contain a negative range, for instance, generalized skewed-t distribution and skewed-t Copulas. Thus, our findings are useful for academics, practitioners, and policy makers.

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Statistics and applied statistics

Briš¹,

Dao²

2016

Applied Mathematics in Engineering and Reliability

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On the distortion risk measure using copulas

Cited by 4 publications

References 6 publications

Determining Distribution for the Product of Random Variables by Using Copulas

Determining Distribution for the Product of Random Variables by Using Copulas

Determining Distribution for the Quotients of Dependent and Independent Random Variables by Using Copulas

Statistics and applied statistics

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