Multivariate double truncated expectation and covariance risk measures for elliptical distributions

TL;DR

This paper derives exact formulas for multivariate and univariate double truncated expectation and covariance measures within elliptical distributions, with applications to financial industry stock returns.

Contribution

It provides new explicit expressions for MDTE and MDTCov for elliptical distributions, including special cases like normal and Student-t distributions.

Findings

Exact formulas for MDTE and MDTCov derived for several elliptical distributions.

Numerical calculations of DTE, DTV, MDTE, and MDTCov for normal distribution.

Application to stock return data from financial industry segments.

Abstract

The main objective of this work is to calculate the multivariate double truncated expectation (MDTE) and covariance (MDTCov) for elliptical distributions. We also consider double truncated expectation (DTE) and variance (DTV) for univariate elliptical distributions. The exact expressions of MDTE and MDTCov are derived for some special cases of the family, such as normal, student-$t$, logistic, Laplace and Pearson type VII distributions. As numerical illustration, the DTE, DTV, MDTE and MDTCov for normal distribution are computed in details. Finally, we discuss MDTE and MDTCov of three industry segments' (Banks, Insurance, Financial and Credit Service) stock return in London stock exchange.