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Risk measures for multivariate financial positions are studied in a utility-based framework. Under a certain incomplete preference relation, shortfall and divergence risk measures are defined as the optimal values of specific set…
Marginal expected shortfall is unquestionably one of the most popular systemic risk measures. Studying its extreme behaviour is particularly relevant for risk protection against severe global financial market downturns. In this context,…
It is shown that the axioms for coherent risk measures imply that whenever there is an asset in a portfolio that dominates the others in a given sample (which happens with finite probability even for large samples), then this portfolio…
Expected Shortfall (ES, also known as CVaR) is the most important coherent risk measure in finance, insurance, risk management, and engineering. Recently, Wang and Zitikis (2021) put forward four economic axioms for portfolio risk…
Our primary aim is to find an estimate of the expected shortfall in various situations: (1) Nonparametric situation, when the probability distribution of the incurred loss is unknown, only satisfying some general conditions. Then, following…
We present a general framework for a comparative theory of variability measures, with a particular focus on the recently introduced one-parameter families of inter-Expected Shortfall differences and inter-expectile differences, that are…
In this note, we comment on the relevance of elicitability for backtesting risk measure estimates. In particular, we propose the use of Diebold-Mariano tests, and show how they can be implemented for Expected Shortfall (ES), based on the…
This paper investigates risk measures derived from the expected maximum deficit in a continuous-time framework and develops optimal reserve allocation strategies across multiple lines of business. We formalize the expected maximum deficit…
Maximum drawdown, the largest cumulative loss from peak to trough, is one of the most widely used indicators of risk in the fund management industry, but one of the least developed in the context of measures of risk. We formalize drawdown…
We study issues of robustness in the context of Quantitative Risk Management and Optimization. We develop a general methodology for determining whether a given risk measurement related optimization problem is robust, which we call…
We study the design of an optimal insurance contract in which the insured maximizes her expected utility and the insurer limits the variance of his risk exposure while maintaining the principle of indemnity and charging the premium…
Historical (Stressed-) Value-at-Risk ((S)VAR), and Expected Shortfall (ES), are widely used risk measures in regulatory capital and Initial Margin, i.e. funding, computations. However, whilst the definitions of VAR and ES are unambiguous,…
This paper attempts to provide a decision-theoretic foundation for the measurement of economic tail risk, which is not only closely related to utility theory but also relevant to statistical model uncertainty. The main result is that the…
We introduce a novel regression framework which simultaneously models the quantile and the Expected Shortfall (ES) of a response variable given a set of covariates. This regression is based on a strictly consistent loss function for the…
The debate of what quantitative risk measure to choose in practice has mainly focused on the dichotomy between Value at Risk (VaR) -- a quantile -- and Expected Shortfall (ES) -- a tail expectation. Range Value at Risk (RVaR) is a natural…
The Effective Sample Size (ESS) is an important measure of efficiency of Monte Carlo methods such as Markov Chain Monte Carlo (MCMC) and Importance Sampling (IS) techniques. In the IS context, an approximation $\widehat{ESS}$ of the…
This paper discusses an alternative explanation for the empirical findings contradicting the positive relationship between risk (variance) and reward (expected return). We show that these contradicting results might be due to the false…
The expectile can be considered as a generalization of quantile. While expected shortfall is a quantile based risk measure, we study its counterpart -- the expectile based expected shortfall -- where expectile takes the place of quantile.…
We study a general risk measure called the generalized shortfall risk measure, which was first introduced in Mao and Cai (2018). It is proposed under the rank-dependent expected utility framework, or equivalently induced from the cumulative…
In the last five years, expected shortfall (ES) and stressed ES (SES) have become key required regulatory measures of market risk in the banking sector, especially following events such as the global financial crisis. Thus, finding ways to…