Generalized Johnson Distributions and Risk Functionals

Abstract

In this paper, we study the generalized Johnson distributions’ class and its applications in finance and risk theory. The recent literature on Johnson distributions displays a better gooodness of fitting for data coming from financial markets, such as portfolio returns. However, a gereral question in risk theory and finance is the following: Which class of distributions is more appropriate in order to determine the behaviour of data coming from financial markets and insurance claims? Another question is the following one: Is ther any class of distributions that is appropriate for calculations related to any kind of risk faced by financial isntitutions and insurance companies? The answer proposed to these questions is the use of generalized Johnson’s distributions. The parameters of such distributions are estimated by the order statistics of a single or more samples. Risk functionals represent a unified approach comprising every kind of risk metric. Risk functionals include value-at-risk and expected shortfall, coherent risk measures, and endpoints and thresholds. We deduce that the risk functionals sastisfy convexity—like properties with respect to finitely-mixed distributions. We also prove in detail that the empirical distribution is a reasonable way for the estimation of the above risk functionals. In the Appendix, we provide two numerical examples for fitting samples of portfolio returns under the Johnson’s transformation.