Evaluating Range Value at Risk Forecasts

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 interpolation between these two prominent risk measures, which constitutes a tradeoff between the sensitivity of the latter and the robustness of the former, turning it into a practically relevant risk measure on its own. As such, there is a need to statistically validate RVaR forecasts and to compare and rank the performance of different RVaR models, tasks subsumed under the term 'backtesting' in finance. The predictive performance is best evaluated and compared in terms of strictly consistent loss or scoring functions. That is, functions which are minimised in expectation by the correct RVaR forecast. Much like ES, it has been shown recently that RVaR does not admit strictly consistent scoring functions, i.e., it is not elicitable. Mitigating this negative result, this paper shows that a triplet of RVaR with two VaR components at different levels is elicitable. We characterise the class of strictly consistent scoring functions for this triplet. Additional properties of these scoring functions are examined, including the diagnostic tool of Murphy diagrams. The results are illustrated with a simulation study, and we put our approach in perspective with respect to the classical approach of trimmed least squares in robust regression.Comment: 25 pages, 2 figures An earlier version of this paper was circulated under the name 'Elicitability of Range Value at Risk'. The presentation has been made more concise and minor errors have been corrected. Statistics & Risk Modeling, 202

Elicitability of range value at risk

The predictive performance of point forecasts for a statistical functional, such as the mean, a quantile, or a certain risk measure, is commonly assessed in terms of scoring (or loss) functions. A scoring functions should be (strictly) consistent for the functional of interest, that is, the expected score should be minimised by the correctly specified functional value. A functional is elicitable if it possesses a strictly consistent scoring function. In quantitative risk management, the elicitability of a risk measure is closely related to comparative backtesting procedures. As such, it has gained considerable interest in the debate about which risk measure to choose in practice. While this discussion has mainly focused on the dichotomy between Value at Risk (VaR) - a quantile - and Expected Shortfall (ES) - a tail expectation, this paper is concerned with Range Value at Risk (RVaR). RVaR can be regarded as an interpolation of VaR and ES, which constitutes a tradeoff between the sensitivity of the latter and the robustness of the former. Recalling that RVaR is not elicitable, we show that a triplet of RVaR with two VaR components at different levels is elicitable. We characterise the class of strictly consistent scoring functions. Moreover, additional properties of these scoring functions are examined, including the diagnostic tool of Murphy diagrams. The results are illustrated with a simulation study, and we put our approach in perspective with respect to the classical approach of trimmed least squares in robust regression

Advancing stability analysis of mean-risk stochastic programs: Bilevel and two-stage models

Measuring and managing risk has become crucial in modern decision making under stochastic uncertainty. In two-stage stochastic programming, mean-risk models are essentially defined by a parametric recourse problem and a quantification of risk. The thesis addresses sufficient conditions for weak continuity of the resulting objective functions with respect to perturbations of the underlying probability measure. The approach is based on so called psi-weak topologies that are finer than the topology of weak convergence and allows to unify and extend known results for a comprehensive class of risk measures and recourse problems. In particular, stability of mean-risk models with mixed-integer quadratic and general mixed-integer convex recourse problems is derived for any law-invariant, convex and nondecreasing quantification of risk. From a conceptual point of view, two-stage stochastic programs and bilevel problems under stochastic uncertainty are closely related. Assuming that only the follower can observe the realization of the randomness, the optimistic and pessimistic setting give rise to two-stage problems where only optimal solutions of the lower level are feasible for the recourse problem. So far, stability in stochastic bilevel programming has only been examined for a specific model based on a quantile criterion. The novel approach allows to identify sufficient conditions for stability of stochastic bilevel problems with quadratic lower level and is applicable for a comprehensive class of risk measures.Die Bewertung und das Management von Risken sind ein wesentlicher Aspekt von Entscheidungsproblemen unter stochastischer Unsicherheit. Zielfunktionsbasierte risikoaverse Modelle der zweistufigen stochastischen Optimierung lassen sich im Wesentlichen durch ihr parametrisches Zweitstufenproblem und das betrachtete Risikomaß charakterisieren. Die Arbeit befasst sich mit hinreichenden Bedingungen für Stetigkeit der resultierenden Zielfunktion unter Störungen des zu Grunde liegenden Wahrscheinlichkeitsmaßes bezüglich der Topologie schwacher Konvergenz. Der Ansatz basiert auf so genannten psi-schwachen Topologien, die feiner als die Topologie schwacher Konvergenz sind. Für eine umfassende Klasse von Risikomaßen und Zweitstufenproblemen werden so bestehende Resultate vereinheitlicht und erweitert. Insbesondere lassen sich für jedes verteilungsinvariante, konvexe und nichtfallende Risikomaß Stabilitätsaussagen für Aufgaben mit quadratischem oder konvexem gemischt-ganzzahligen Zweitstufenproblem treffen. Aus konzeptioneller Sicht sind zweistufige stochastische Programme und Bilevel Probleme unter stochastischer Unsicherheit eng miteinander verbunden. Unter der Annnahme, dass nur der Entscheider auf der unteren Ebene die Realisierung des Zufalls beobachten kann, führen sowohl der optimistische als auch der pessimistische Ansatz auf ein zweistufiges stochastisches Programm. Bei diesem sind nur die Optimallösungen der unteren Ebene zulässig für das Zweitstufenproblem. Bisher ist die Stabilität solcher Aufgaben nur für Modelle mit einem speziellen Quantilkriterium untersucht worden. Der neue Ansatz erlaubt es, hinreichende Bedingungen für die Stabilität von stochastischen Bilevel Problemen mit quadratischem Nachfolgerproblem zu identifizieren und ist auf eine reichhaltige Klasse von Risikomaßen anwendbar

Risk Sharing and Risk Aggregation via Risk Measures

Risk measures have been extensively studied in actuarial science in the guise of premium calculation principles for more than 40 years, and recently, they have been the standard tool for financial institutions in both calculating regulatory capital requirement and internal risk management. This thesis focuses on two topics: risk sharing and risk aggregation via risk measures. The problem of risk sharing concerns the redistribution of a total risk among agents using risk measures to quantify risks. Risk aggregation is to study the worst-case value of aggregate risks over all possible dependence structures with given marginal risks. On the first topic, we address the problem of risk sharing among agents using a two-parameter class of quantile-based risk measures, the so-called Range-Value-at-Risk (RVaR), as their preferences. The family of RVaR includes the Value-at-Risk (VaR) and the Expected Shortfall (ES), the two popular and competing regulatory risk measures, as special cases. We first establish an inequality for RVaR-based risk aggregation, showing that RVaR satisfies a special form of subadditivity. Then, the Pareto-optimal risk sharing problem is solved through explicit construction. We also study risk sharing in a competitive market and obtain an explicit Arrow-Debreu equilibrium. Robustness and comonotonicity of optimal allocations are investigated, and several novel advantages of ES over VaR from the perspective of a regulator are revealed. Reinsurance, as a special type of risk sharing, has been studied extensively from the perspective of either an insurer or a reinsurer. To take the interests of both parties into consideration, we study Pareto optimality of reinsurance arrangements under general model settings. We give the necessary and sufficient conditions for a reinsurance contract to be Pareto-optimal and characterize all such optimal contracts under more general model assumptions. Sufficient conditions that guarantee the existence of the Pareto-optimal contracts are obtained. When the losses of an insurer and a reinsurer are measured by the ES risk measures, we obtain the explicit forms of the Pareto-optimal reinsurance contracts under the expected value premium principle. On the second topic, we first study the aggregation of inhomogeneous risks with a special type of model uncertainty, called dependence uncertainty, in individual risk models. We establish general asymptotic equivalence results for the classes of distortion risk measures and convex risk measures under different mild conditions. The results implicitly suggest that it is only reasonable to implement a coherent risk measure for the aggregation of a large number of risks with dependence uncertainty. Then, we bring the well studied dependence uncertainty in individual risk models into collective risk models. We study the worst-case values of the VaR and the ES of the aggregate loss with identically distributed individual losses, under two settings of dependence uncertainty: (i) the counting random variable and the individual losses are independent, and the dependence of the individual losses is unknown; (ii) the dependence of the counting random variable and the individual losses is unknown. Analytical results for the worst-case values of ES are obtained. For the loss from a large portfolio of insurance policies, the asymptotic equivalence of VaR and ES is established, and approximation errors are obtained under the two dependence settings