Risk management with risk measures

The series of working papers contains recent advances on the theory of risk measures and their applications in risk management. Different from the series Axiomatic theory of risk measures, papers in this series focus on important and practical issues such as optimization, inference, computation, capital allocation, robustness, and model uncertainty. Many papers pay special attention to Value-at-Risk (VaR) and the Expected Shortfall (ES, also known as TVaR, CVaR and AVaR), which are widely used in global financial regulation and internal risk management. Some particular contributions are:

A brief description of each working paper is provided below to explain its main results and the logical structure across papers.

We introduce the class of monotonic mean-deviation risk measures, giving rise to many new explicit examples of convex and nonconvex consistent risk measures.

This is a follow up of WP14 on DQ. The main focus is DQ constructed from VaR and ES and its formulas for elliptical and multivariate regularly varying models. The journal version is published in Insurance: Mathematics and Economics (2023).

We introduce the diversification quotient (DQ) as a new diversification index based on risk measures, and develop its theory. DQ has many advantages over existing indices: it satisfies many nice technical properties that are not satisfied by competing concepts; it is efficient to optimize in portfolio selection; it can properly capture tail heaviness and common shocks; it is intuitive to interpret, and its empirical performance is competitive. The journal version is to appear in Management Science (2024).

This paper is a follow-up of WP05 on PELVE. We study the the problem of how to obtain a distribution from a given PELVE curve, or some partial information. A notable finding is that the PELVE curve does not determine the distribution up to a location-scale family. The journal version is published in North American Actuarial Journal (2024).

Inspired by the celebrated ES optimization formula of Rockafellar and Uryasev, this paper provides a reverse ES optimization formula, which says that a mean excess function at any fixed threshold is the maximum of an ES curve minus a linear function. The reverse ES optimization formula is closely related to the Fenchel-Legendre transforms and helps to solve risk evaluation problems under model uncertainty. The journal version is to appear in North American Actuarial Journal (2023).
- After the paper's publication, we became aware of the existence of earlier results very similar to the reverse ES formula in Theorem 3.1 of Ogryczak and Ruszczyński (2002) and Theorem 2 of Rockafellar and Royset (2014). They should have been given credit.

We introduce a new approach for prudent risk evaluation based on stochastic dominance, which will be called the model aggregation (MA) approach. In contrast to the classic worst-case risk (WR) approach, the MA approach produces not only a robust value of risk evaluation but also a robust distributional model, and it has many advantages in optimization. Equivalence properties between the MA and the WR approaches leads to new axiomatic characterizations for popular risk measures including VaR, ES, and adjusted ES (WP06). The MA approach for Wasserstein and mean-variance uncertainty sets admits explicit formulas for the obtained robust models.

We present a comprehensive theoretical framework for stressing mechanisms which are designed to obtain stressed scenarios for multivariate stochastic models with a focus on dependence among components in the model. We propose and study in detail two families of stressing mechanisms, based respectively on mixtures of univariate stresses and on transformations of statistics we call Spearman and Kendall's cores. The journal version is to appear in European Journal of Operational Research (2024).

We characterize a few classes of risk measures, including ES, from the perspective of optimal insurance design. This consideration is very different from most papers in the series. The main result is that if the Pareto-efficient contracts have a deductible form, then the induced risk measures of the insured and the insurer are both a mixture of ES and the mean. The journal version is published in Insurance: Mathematics and Economics (2022).

This paper provides a general framework for variability measures. The one-parameter families of inter-quantile, inter-ES, and inter-expectile differences are studied in detail. The journal version is published in Insurance: Mathematics and Economics (2022).

We establish a unifying equivalence result which converts a non-convex distortion riskmetric (including distortion risk measures) under distributional uncertainty to a convex one. This result needs the new notion of closedness under concentration. The results can be applied to portfolio optimization, optimization under moment constraints, and preference robust optimization. The journal version is to appear in Mathematical Programming (2024).

We introduce the class of Adjusted Expected Shortfall, a new class of convex risk measures, including the classic ES. This class of risk measures is intimately linked to second-order stochastic dominance. The journal version is published in Journal of Banking and Finance (2022).

A new distributional index, the probability equivalence level of VaR and ES (PELVE) is introduced. Among several theoretical features, PELVE distinguishes heavy-tailed distributions from light-tailed ones via a threshold of approximately 2.72. Applying PELVE to financial data, we observe an empirical evidence that the use of ES rewards portfolio diversification more than the use of VaR. The journal version is published in Journal of Econometrics (2023).

A new notion of robustness against optimization is proposed, which is specifically designed for regulatory risk measures, and is conceptually different from robust optimization. For general optimization settings, VaR leads to non-robust optimizers whereas convex risk measures, such as ES, leads to robust ones. The journal version is published in Operations Research (2022).

This paper provides a statistical treatment of capital allocation and sensitivity anlysis. A new ES-based VaR allocation method is introduced to overcome the practical challenges in estimating the classic Euler VaR allocation. The journal version is published in Mathematical Finance (2019).
- Errata in Theorem 3.1 of the journal version: 1) the first p* in the denominator of the second term in (9) should be p; 2) the convergence type in this theorem should all be in distribution (not in probability).

The Gini Shortfall (GS) is introduced and its theoretical properties and applications are explored. For suitable values of its parameters, GS is a coherent risk measure which is a combination of ES and the tail Gini variation. The journal version is published in Journal of Banking and Finance (2017).

Subadditivity is the key property which distinguishes VaR and ES. This paper contains seven (old or new) proofs of the subadditivity of ES, which can be used for an educational purpose in quantitative risk management. The journal version is published in Dependence Modeling (2015).