Thetcopula with multiple parameters of degrees of freedom: bivariate characteristics and application to risk management

Abstract

The t copula is often used in risk management as it allows for modelling tail dependence between risks and it is simple to simulate and calibrate. However, the use of a standard t copula is often criticized due to its restriction of having a single parameter for the degrees of freedom (dof) that may limit its capability to model the tail dependence structure in a multivariate case. To overcome this problem, grouped t copula was proposed recently, where risks are grouped a priori in such a way that each group has a standard t copula with its specific dof parameter. In this paper we propose the use of a grouped t copula, where each group consists of one risk factor only, so that a priori grouping is not required. The copula characteristics in the bivariate case are studied. We explain simulation and calibration procedures, including a simulation study on finite sample properties of the maximum likelihood estimators and Kendall's tau approximation. This new copula can be significantly different from the standard t copula in terms of risk measures such as tail dependence, value at risk and expected shortfall.

Robust calibration of hierarchical population models for heterogeneous cell populations

“…While the Student's t distribution suffered from problems of over-fitting in the case of population-average data (Maier et al, 2017), the number of measurements in single-cell data sets is usually much higher and we do not expect to face the same problems as for population-average data. To allow for different degrees of freedom in multivariate measurements, a t copula could be employed (Luo and Shevchenko, 2010). Also, a skewed version of the Student's t distribution as, e.g., used by Pyne et al (2009) could be incorporated.…”

Anti-robust and tonsured statistics

“…Tail dependence is another non-parametric technique, which is complementary to tonsuring. In two dimensions, upper right tail dependence λ ur (u), as u goes to zero, is defined for any bivariate copula density c(x,y) as (9) where changing u to 1-u and > to < in the obvious way gets the formulae for upper left (ul), bottom right (br), and bottom left (bl) tail dependencies. In words, this is a measure of whether copula density for a small square of size u 2 at the corner ( angle z/||z|| ) at 45˚, 135˚, 225˚, or 315˚ stays finite, or goes to zero, as the square gets smaller.…”

Comparison analysis of copula-based and Markowitz portfolio methods

“…The concept of copula is based on Shklar theorem [9]. In [10][11][12][13][14] they can conclude that since 2007-2008 elliptical copulas become popular for the purpose of mar ket risk assessment. However, Gaussian copula ignores tail dependence.…”