Using equations that arise in quantum mechanics, this paper describes a way to more accurately and efficiently represent non-Gaussian return distributions than the standard method of invoking skewness and kurtosis. Then, it provides a new single intuitive number, defined here as the “crash...

We revisit mean-risk portfolio selection in a one-period financial market where risk is quantified by a positively homogeneous risk measure ρ on L1. We first show that under mild assumptions, the set of optimal portfolios for a fixed return is nonempty and compact. However, unlike in classical...

We introduce and study the main properties of a class of convex risk measures that refine Expected Shortfall by simultaneously controlling the expected losses associated with different portions of the tail distribution. The corresponding adjusted Expected Shortfalls quantify risk as the minimum...

This paper studies mean-risk portfolio selection in a one-period financial market, where risk is quantified by a star-shaped risk measure ρ. We introduce two new axioms: weak and strong sensitivity to large losses. We show that the first axiom is key to ensure the existence of optimal...

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...

The paper deals with maritime risk, which we consider important, no doubt, for ship-owners acting in volatile markets. Traditionally, risk is measured by "standard deviation". Other risk measures like "excess kurtosis", "excess skewness", "long-term dependence" and the "catastrophe propensity"...