Conditional Expectile: An Alternative to Value at Risk (VaR)

Abstract

Various risk measures have been reviewed against the criteria commonly accepted by financial researchers and practitioners: coherence, elicitability, comonotonic additivity, and intuitiveness. It follows that the only risk measure that is both coherent and elicitable is an Expectile based risk measure. But unlike the VaR measure, the Expectile does not meet the criterion of comonotonic additivity. Taking this result into account, this paper proposes Expectile conditioned by a VaR as a risk measure. Then, the principle of elicitability has been used to perform backtesting and comparative analysis of the performance of the new measure versus VaR and Expectile. Two kinds of conditioning have been retained, Expectile conditioned to a predictive VaR (CEVaR) and Expectile conditioned to a realized VaR (CRVaR) as alternative risk measures. The Monte Carlo simulation and historical simulation methods were implemented to assess the validity of the two risk measures CEVaR and CRVaR. The evaluation was carried out in terms of: - Precision: for this purpose, the backtesting functions associated with the risk measures have been used; - Sub-additivity and comonotonic additivity. The simulations showed that the CRVaR and CEVaR measures are more accurate than the Expectile and VaR measures but CEVaR is not stable when the historical simulation method is used. A numerical assessment of sub-additivity and comonotonic additivity showed that CRVaR is sub-additive while CEVAR is not and both are not comonotonic additive. Nevertheless, we could conclude that CRVAR is a valid alternative to VaR as it’s coherent and outperforms Expetile in terms of accuracy.