Expectile Linked Expected Shortfall Bounds

Downside risk measures play a very interesting role in Actuarial Science and Mathematical Finance. In particular, the value at risk (VaR) and the expected shortfall (ES) have become very important instruments in order to address risk management problems, capital requirements, portfolio selection, optimal reinsurance, pricing and hedging issues, risk transference, risk sharing, etc. In contrast, expectile risk measures are not as widely used, though they are both coherent and elicitable. This paper shows another interesting property affecting expectiles. Indeed, expectiles enable us to give VaR and ES upper bounds applying to every level of confidence. In other words, expectiles allow us to control VaR and ES regardless of the level of confidence. In general, once an actuarial or financial strategy is selected by dealing with VaR or ES with a specific confidence level, nothing is known about the effectiveness of such a strategy if the confidence level becomes different. Nevertheless, the use of expectiles may overcome this caveat. Illustrative actuarial applications will be presented