• The Korean Journal of Applied Statistics
• /
• v.24 no.5
• /
• pp.809-820
• /
• 2011

VaR(Value at risk) is a measure of market risk management and needs to be estimated for multiple distributions. In this paper, Copula functions are used to generate distributions of multivariate random variables. The dependence structure of random variables is classified by the exchangeable Copula, fully nested Copula, partially nested Copula. For the earning rate data of four Korean industries, the parameters of the Archimedean Copula functions including Clayton, Gumbel and Frank Copula are estimated by using three kinds of dependence structure. These Copula functions are then fitted to to the data so that corresponding VaR are obtained and explored.

• Journal of the Korean Data and Information Science Society
• /
• v.28 no.2
• /
• pp.421-433
• /
• 2017

The CTE (conditional tail expectation) is a useful risk management measure for a diversified investment portfolio that can be generally estimated by using a transformed univariate distribution. Hong et al. (2016) proposed a multivariate CTE based on multivariate quantile vectors, and explored its characteristics for multivariate normal distributions. Since most real financial data is not distributed symmetrically, it is problematic to apply the CTE to normal distributions. In order to obtain a multivariate CTE for various kinds of joint distributions, distribution fitting methods using copula functions are proposed in this work. Among the many copula functions, the Clayton, Frank, and Gumbel functions are considered, and the multivariate CTEs are obtained by using their generator functions and parameters. These CTEs are compared with CTEs obtained using other distribution functions. The characteristics of the multivariate CTEs are discussed, as are the properties of the distribution functions and their corresponding accuracy. Finally, conclusions are derived and presented with illustrative examples.

• The Korean Journal of Applied Statistics
• /
• v.24 no.3
• /
• pp.523-533
• /
• 2011

Most nancial preference methods for market risk management are to estimate VaR. In many real cases, it happens to obtain the VaRs of the univariate as well as multivariate distributions based on multivariate data. Copula functions are used to explore the dependence of non-normal random variables and generate the corresponding multivariate distribution functions in this work. We estimate Archimedian Copula functions including Clayton Copula, Gumbel Copula, Frank Copula that are tted to the multivariate earning rate distribution, and then obtain their VaRs. With these Copula functions, we estimate the VaRs of both a certain integrated industry and individual industries. The parameters of three kinds of Copula functions are estimated for an illustrated stock data of two Korean industries to obtain the VaR of the bivariate distribution and those of the corresponding univariate distributions. These VaRs are compared with those obtained from other methods to discuss the accuracy of the estimations.

• Proceedings of the Korea Water Resources Association Conference
• /
• 2016.05a
• /
• pp.229-229
• /
• 2016

현재 국내외에서 제공되고 있는 기후변화 시나리오 자료의 경우 일단위로 제공되고 있다. 그러나 수자원 설계 및 계획 시 중요한 입력자료 중 하나는 시간단위 강우 자료이다. 이러한 시간단위 자료는 강우-유추 분석, 댐 설계 및 위험도 분석에 있어 중요한 입력 변수중 하나이므로 기후변화 시나리오에 따른 영향을 평가하기 위해선 신뢰성 있는 상세화 기법이 필요하다. 국내외에서는 일단위에서 일단위로 상세화 하는 기법, 또는 공간상세화 기법 연구는 상당히 진행된바 있는 반면, 시간단위 상세화 기법 연구는 일단위 연구에 비해 상대적으로 미진한 실정이다. 즉 일단위 상세화 기법의 경우 Weather generator, Weather typing 등 다양한 기법이 존재하고 이를 활용한 연구사례가 많지만, 시간단위 상세화 기법의 Poisson 기법을 활용한 사례가 다수 존재하였다. 이러한 이유로 본 연구에서는 기후변화 시나리오에 따른 영향을 평가하기 위해 Bayesian 기법을 도입하여 신뢰성 있는 시간단위 강우량을 생성할 수 있는 모형을 개발하였으며, 연대별로 산정된 결과는 빈도해석을 통해 미래 확률강우량을 제시하였다. 본 연구에서 제안하고자 하는 Bayesian Copula 모형은 기존 주변확률분포(marginal distribution) 매개변수와 Copula 매개변수 추정시 각각 다른 기법을 활용하여 추정하며, 각각 모형에서 발생하는 불확실성은 추정하지 못하는 반면, Bayesian Copula 모형의 경우 매개변수의 사후분포를 정량적으로 제시할 수 있으며, 추정되는 확률강우량 역시 불확실성을 정량적으로 산정할 수 있는 장점을 확인할 수 있었다.

• Bulletin of the Korean Mathematical Society
• /
• v.54 no.3
• /
• pp.839-874
• /
• 2017

Our goal is to establish and prove the almost sure central limit theorems for some order statistics $\{M_n^{(k)}\}$, $k=1,2,{\ldots}$, formed by stochastic processes ($X_1,X_2,{\ldots},X_n$), $n{\in}N$, the distributions of which are defined by certain Archimedean copulas. Some properties of generators of such the copulas are intensively used in our proofs. The first class of theorems stated and proved in the paper concerns sequences of ordinary maxima $\{M_n\}$, the second class of the presented results and proofs applies for sequences of the second largest maxima $\{M_n^{(2)}\}$ and the third (and the last) part of our investigations is devoted to the proofs of the almost sure central limit theorems for the k-th largest maxima $\{M_n^{(k)}\}$ in general. The assumptions imposed in the first two of the mentioned groups of claims significantly differ from the conditions used in the last - the most general - case.