Conditional Coherent and Convex Risk Measures Under Uncertainty

We consider modeling and implementation of risk-averse preferences in stochastic programming problems using axiomatically defined risk measures. We derive representations for several classes of risk measures (e.g., coherent risk measures, deviation measures) via solutions of specially formulated stochastic programming problems that facilitate incorporation of risk measures in multistage stochastic programming problems. As an illustration of the general approach, we consider a two-stage stochastic weapon-target assignment problem, where a coherent risk measure is used to capture the risk of the second-stage (recourse) action.

Probabilistically distorted risk-sensitive infinite-horizon dynamic programming

Regulatory Arbitrage of Risk Measures

Despite the fact that the value-at-risk (VaR) measure has been adopted as a standard risk measure in the financial industry, it has a number of deficiencies recognized by financial professionals. It is not a coherent risk measure. This is because it does not satisfy the subadditivity property requirement of a coherent risk measure. That is, there are cases in which the portfolio VaR is larger than the sum of the VaRs of the portfolio constituents, supporting the view that VaR cannot be used as a true risk measure. Unlike VaR, the average value-at-risk measure (AVaR)—also referred to as conditional value-at-risk and expected shortfall—is a coherent risk measure and has other advantages that result in its greater acceptance in risk modeling. Keywords: average value-at-risk; coherent risk measures; average value-at-risk; conditional value-at-risk; expected shortfall; The Student's; distribution

Coherent risk measures have been introduced by Artzner et al. in financial markets where the set of states Ω is finite. This has been extended by Delbaen to cases where Ω is infinite. The financial markets are constantly evolving and the classical definition of risk measures is static. Hence, the need to study its complexity and evolution. Complexity theory focuses on the interdisciplinary study of its systems. In its evolution, the study of complex adaptive systems and the emergent phenomena associated with them are of fundamental importance. In this paper, we consider the coherent risk measure and their interpretation as complex adaptive system. KEYWORD: Coherent risk measure; complex adaptive system.

Measuring and managing the risk of extensive distribution network outages during extreme events is critical for ensuring system-level energy balance in transmission network operations. However, existing risk measures used in stochastic optimization of power systems are computationally intractable for this problem involving large numbers of discrete random variables. Using a new coherent risk measure, Entropic Value-at-Risk (EVaR), that requires significantly less computational complexity, we propose an EVaR-constrained optimal power flow model that can quantify and manage the outage risk of extensive distribution feeders. The optimization problem with EVaR constraints on discrete random variables is equivalently reformulated as a conic programming model, which allows the problem to leverage the computational efficiency of conic solvers. The superiority of the proposed model is validated on the real-world Puerto Rico transmission grid combined with its large-scale distribution networks.

The Basel Committee of Banking Supervision has used value-at-risk as a measure of market risk in the trading book for two decades in several accords. After the global financial crisis of 2008-2009 value-at-risk was criticised on the grounds that as a measure of risk is not sub-additive, in the sense that it behaves very erratically when banks or regulators try to aggregate compartmentalised risk across all branches of a large diverse bank. Expected shortfall emerged as a natural alternative of value-at-risk fulfilling all four axioms of a coherent risk measure and belongs to the category of spectral risk measures which are not elicitable unless they reduce to minus the expected value. Another critical issue is that the Basel committee indicates that risk managers should conduct the risk measurement tests based on a dataset which is very small for high confidence levels calculations. In this work we calculate and apply the theoretical tail conditional expectation of the standardised Pearson type IV distribution. This is a coherent measure of risk and assuming it describes appropriately the data generation process it can provide the risk manager with reliable results.

In this paper, we provide a new representation result for dynamic capital allocations and dynamic convex risk measures that are based on backward stochastic differential equations (BSDEs). We derive this representation from a classical differentiability result for BSDEs and the full allocation property of the Aumann–Shapley allocation. The representation covers BSDE-based dynamic convex and dynamic coherent risk measures. The results are applied to derive a representation for the dynamic entropic risk measure. Our results are also applicable in a specific way to the static case, where we are able to derive a new representation result for static convex risk measures that are Gâteaux-differentiable.

A spectrum of upper bounds (Qα(X ; p) αε[0,∞] on the (largest) (1-p)-quantile Q(X;p) of an arbitrary random variable X is introduced and shown to be stable and monotonic in α, p, and X , with Q0(X ;p) = Q(X;p). If p is small enough and the distribution of X is regular enough, then Qα(X ; p) is rather close to Q(X ; p). Moreover, these quantile bounds are coherent measures of risk. Furthermore, Qα(X ; p) is the optimal value in a certain minimization problem, the minimizers in which are described in detail. This allows of a comparatively easy incorporation of these bounds into more specialized optimization problems. In finance, Q0(X;p) and Q1(X ; p) are known as the value at risk (VaR) and the conditional value at risk (CVaR). The bounds Qα(X ; p) can also be used as measures of economic inequality. The spectrum parameter α plays the role of an index of sensitivity to risk. The problems of the effective computation of the bounds are considered. Various other related results are obtained.

A standard deviation has been a starting point for a mathematical definition of risk. As a remedy for drawbacks such as subadditivity property discouraging the diversification, coherent and convex risk measures are introduced in an axiomatic approach. Choquet expectation and g-expectations, which generalize mathematical expectations, are widely used in hedging and pricing contingent claims in incomplete markets. The each risk measure or expectation give rise to its own pricing rules. In this paper we investigate relationships among dynamic risk measures, Choquet expectation and dynamic g-expectations in the framework of the continuous-time asset pricing.

In this paper, based on the entropy which is used as a measure of the uncertainty, a risk measure of the Intellectual Property Risk is constructed via using the coherent risk measure of finance risk for reference. To construct the intellectual property risk measure, we define the intellectual property risk as a risk composed of three elements, namely uncertainty, variance and loss. The risk measure constructed in this paper is proved to be able to reflect the people's perception of intellectual property risk and is a coherent risk measure.

Asymptotic Equivalence of Risk Measures Under Dependence Uncertainty

We show how the dynamic optimization problem with the capital constraint can be reduced to the problem to find an optimal modified claim where is a randomized test in the static problem. Coherent risk measure is used as risk measure in the random variable spaces. The paper is written in expository style to some degree. We use an average risk of measure(AVaR), which is a special coherent risk measure, to see how to hedge the modified claim in a complete market model.

Closed-Form Solutions for Worst-Case Law Invariant Risk Measures with Application to Robust Portfolio Optimization

A note on the Swiss Solvency Test risk measure