We present the Shortfall Deviation Risk (SDR), a risk measure that represents the expected loss that occurs with certain probability penalized by the dispersion of results that are worse than such an expectation. SDR combines Expected Shortfall (ES) and Shortfall Deviation (SD), which we also...

In this paper we propose an expected shortfall (ES) backtesting approach that uses the dispersion of a truncated distribution by the estimated value-at-risk (VaR) upper limit, does not limit the approach to the Gaussian case and allows us to test if each individual VaR violation is significantly...

Ever since the introduction of Markowitz's classical quadratic programming problem, transforming portfolio optimization into a linear programming (LP) problem has drawn much attention from researchers and practitioners, given the tractability of LP. However, using non-linear risk measures and...

Liquidity is easily perceived but not easily defined in financial markets. In this study, we present a class of liquidity measures called Proper Liquidity Measures (PLM). We prove that widely used measure such as percent quoted spreads and the Amihud measure fulfill our axioms. PLMs can be...