A New Family of Expectiles and its Properties

Abstract

Introduction. This paper considers a risk measure called expectile. Expectile is a characteristic of a random variable calculated using the asymmetric least square method. The level of asymmetry is defined by a parameter in the interval (0, 1). Expectile is used in financial applications, portfolio optimization problems, and other applications as well as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). But expectile has a set of advantageous properties. Expectile is both a coherent and elicitable risk measure that takes into account the whole distribution and assigns greater weight to the right tail. The purpose of the paper. As a rule, expectile is compared with quantile (VaR). Our goal is to compare expectile with CVaR by introducing the same parameter – confidence level. To do this we first give a new representation of expectile using the weighted sum of mean and CVaR. Then we consider a new family of expectiles defined by two parameters. Such expectiles are compared with quantile and CVaR for different continuous and finite discrete distributions. Our next goal is to build a regular risk quadrangle where expectile is a risk function. Results. We propose and substantiate two new expressions that define expectile. The first expression uses maximization by varying confidence level of CVaR and varying coefficient before CVaR. It is specified for continuous and finite discrete distributions. The second expression uses minimization of the new error function of the new expectile-based risk quadrangle. The use of two parameters in expectile definition changes the dependence of expectile on its confidence level and generates a new family of expectiles. Comparison of new expectiles with quantile and CVaR for a set of distributions shows that the proposed expectiles can be closer to the quantile than the standard expectile. We propose two variants for expectile linearization and show how to use them with a linear loss function. Keywords: Expectile, EVaR, Quantile, Conditional Value-at-Risk, CVaR, Kusuoka representation, Fundamental Risk Quadrangle, Portfolio Safeguard package.