Differential equations connecting VaR and CVaR

The Value at Risk (VaR) is a very important risk measure for practitioners, supervisors and researchers. Many practitioners draw on VaR as a critical instrument in Risk Management and other Actuarial/Financial problems, while super- visors and regulators must deal with VaR due to the Basel Accords and Solvency II, among other reasons. From a theoretical point of view VaR presents some drawbacks overcome by other risk measures such as the Conditional Value at Risk (CVaR). VaR is neither differentiable nor sub-additive because it is neither continuous nor convex. On the contrary, CVaR satis es all of these properties, and this simpli es many ana- lytical studies if VaR is replaced by CVaR. In this paper several differential equations connecting both VaR and CVaR will be presented. They will allow us to address several important issues involving VaR with the help of the CVaR properties. This new methodology seems to be very efficient. In particular, a new VaR Representation Theorem may be found, and optimization problems involving VaR or probabilistic constraints always have an equivalent differentiable optimization problem. Applications in VaR, marginal VaR, CVaR and marginal CVaR estimates will be addressed as well. An illustrative actuarial numerical example will be given

Skorohod and rough integration for stochastic differential equations driven by Volterra processes

Given a solution Y to a rough differential equation (RDE), a recent result [7] extends the classical Ito-Stratonovich formula and provides a closed-form expression for ∫ Y ○ dX − ∫ Y dX, i.e. the difference between the rough and Skorohod integrals of Y with respect to X, where X is a Gaussian process with finite p-variation less than 3. In this paper, we extend this result to Gaussian processes with finite p-variation such that 3 ≤ p 1/4. As an application we recover Ito formulas in the case where the vector fields of the RDE governing Y are commutative

Stochastic topology optimisation with hierarchical tensor reconstruction

A novel approach for risk-averse structural topology optimization under uncertainties is presented which takes into account random material properties and random forces. For the distribution of material, a phase field approach is employed which allows for arbitrary topological changes during optimization. The state equation is assumed to be a high-dimensional PDE parametrized in a (finite) set of random variables. For the examined case, linearized elasticity with a parametric elasticity tensor is used. Instead of an optimization with respect to the expectation of the involved random fields, for practical purposes it is important to design structures which are also robust in case of events that are not the most frequent. As a common risk-aware measure, the Conditional Value at Risk (CVaR) is used in the cost functional during the minimization procedure. Since the treatment of such high-dimensional problems is a numerically challenging task, a representation in the modern hierarchical tensor train format is proposed. In order to obtain this highly efficient representation of the solution of the random state equation, a tensor completion algorithm is employed which only required the pointwise evaluation of solution realizations. The new method is illustrated with numerical examples and compared with a classical Monte Carlo sampling approach

The Effect of Ratio-based Incentive on Wind Capacity Development and Investment Risk of Wind Units: A System Dynamics Approach

Different capacity incentives like feed-in-tariff have been considered to encourage companies to invest in wind power units. One of the main challenges of the electricity market policymakers is the determination of this fixed payment based on limited funding in a way that the investment cost of wind units is compensated and the associated investment risk is reduced. The main contribution of this paper is the introduction of a method to manage the amount of payment or incentives during a time horizon to reach the targeted wind capacity and reduce its investment risk. In this regard, the ratio-based incentive is introduced. To study the effects of such a policy, the long-term behavior of the electricity market is simulated by a dynamic model, which is a useful tool for policymakers to analyze the effects of their policies. Then, conditional value at risk and value at risk concepts are used to measure the risk of wind capacity investment. The results illustrate that the ratio-based incentive is more effective than the feed-in-tariff in the context of decreasing the risk of investment, reducing total CO2 production, electricity price reduction, and speed of providing higher amounts of wind capacity

Stressing Dynamic Loss Models

Stress testing, and in particular, reverse stress testing, is a prominent exercise in risk management practice. Reverse stress testing, in contrast to (forward) stress testing, aims to find an alternative but plausible model such that under that alternative model, specific adverse stresses (i.e. constraints) are satisfied. Here, we propose a reverse stress testing framework for dynamic models. Specifically, we consider a compound Poisson process over a finite time horizon and stresses composed of expected values of functions applied to the process at the terminal time. We then define the stressed model as the probability measure under which the process satisfies the constraints and which minimizes the Kullback-Leibler divergence to the reference compound Poisson model. We solve this optimization problem, prove existence and uniqueness of the stressed probability measure, and provide a characterization of the Radon-Nikodym derivative from the reference model to the stressed model. We find that under the stressed measure, the intensity and the severity distribution of the process depend on time and the state space. We illustrate the dynamic stress testing by considering stresses on VaR and both VaR and CVaR jointly and provide illustrations of how the stochastic process is altered under these stresses. We generalize the framework to multivariate compound Poisson processes and stresses at times other than the terminal time. We illustrate the applicability of our framework by considering "what if" scenarios, where we answer the question: What is the severity of a stress on a portfolio component at an earlier time such that the aggregate portfolio exceeds a risk threshold at the terminal time? Moreover, for general constraints, we provide a simulation algorithm to simulate sample paths under the stressed measure

Mathematical methods in modern risk measurement: a survey

In the last ten years we have been facing the development on new approaches in Risk Measurement. The Coherent, Expectation Bounded, Convex, Consistent, etc. Risk Measures have been introduced and deeply studied, but there are many open problems that will have to be addressed in forthcoming research. The present paper attempts to summarize the achieved findings and the “State of the Art”, as well as their relationships with other Mathematical Fields, with special focus on other usual topics of Mathematical Finance.En los últimos diez años hemos asistido al desarrollo de nuevos enfoques en Medición de Riesgos. Las Medidas de Riesgo Coherentes, Acotadas por la Media, Convexas, Consistentes, etc., han sido introducidas y profundamente estudiadas, aunque siguen abiertos numerosos problemas que tendrán que ser abordados en investigaciones futuras. El presente artículo sintetiza los logros alcanzados y “El Estado Actual de la Cuestión”, así como las relaciones con otros campos de la Matemática, con atención especial a los temas cl´asicos de la Matem´atica Financiera.This research was partially supported by “Welzia Management SGIIC SA”, “RD Sistemas SA”, “Comunidad Autonoma de Madrid ´ ” (Spain), Grant s-0505/tic/000230, and “MEyC” (Spain), Grant SEJ2006-15401-C04Publicad

Risk measure changes and portfolio optimization theory

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