A note on the Swiss Solvency Test risk measure

In this paper, we take a new perspective to describe the model uncertainty, and thus propose two new classes of risk measures under model uncertainty. To be precise, we use an auxiliary random variable to describe model uncertainty. By proposing new sets of axioms under model uncertainty, we axiomatically introduce and characterize conditional coherent and convex risk measures under a random environment, respectively. As examples, we also discuss the connections of the introduced conditional coherent risk measures under random environments with two existing risk measures. This paper mainly gives some theoretical results, and it is expected to make meaningful complement to the study of coherent and convex risk measures under model uncertainty.

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.

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

Probabilistically distorted risk-sensitive infinite-horizon dynamic programming

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.

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.

Inverse portfolio problem with coherent risk measures

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.

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.

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.

Summary Probabilistic methods for reserves estimation, including uncertainty quantification and probabilistic aggregation, have gained widespread acceptance in the oil and gas industry, since the first comprehensive guidelines were issued by the Society of Petroleum Engineers (SPE) (Swinkels 2001). The probabilistic methods now used in the oil industry, as proposed in these guidelines, are similar to those also used in portfolio theory and risk management by the finance industry. A significant amount can be learned from the extensive experience with probabilistic methods and quantification of risk with measures like value-at-risk (VAR), in financial risk management. Especially, the guidelines issued by the Basel II Accord (Bank for International Settlements 2006) and the discussions since the 2008 financial crisis contain important lessons. In this paper, we examine a fundamental question: "Is the P90 reserves value an appropriate measure for quantifying the reserves’ downside?" For the P90 reserves value to be considered a good measure of the reserves’ downside, it needs to possess a number of basic characteristics involving P90 reserves for each field and the probabilistically aggregated P90 reserves for the portfolio of fields. Analogous to the definition of a coherent risk measure used in the finance industry, we define these characteristics for P90 reserves. The P90 reserves are as good a risk measure as VAR used in the financial industry. However, like VAR, it is not a coherent risk measure. A possible uncertainty scenario, in which one of these necessary characteristics does not hold, is given. An alternative measure of risk for quantifying the reserves’ downside, defined as the average reserves over the confidence interval higher than P90, is presented. This is a coherent risk measure. In this paper, we highlight the appropriateness and limitations of using the P90 reserves estimate as a measure of the reserves’ downside. Understanding of the limitations posed by using the P90 reserves value is vital in management of reserves risk.

Asymptotic Equivalence of Risk Measures Under Dependence Uncertainty

This paper introduces the concept of entropic value-at-risk (EVaR), a new coherent risk measure that corresponds to the tightest possible upper bound obtained from the Chernoff inequality for the value-at-risk (VaR) as well as the conditional value-at-risk (CVaR). We show that a broad class of stochastic optimization problems that are computationally intractable with the CVaR is efficiently solvable when the EVaR is incorporated. We also prove that if two distributions have the same EVaR at all confidence levels, then they are identical at all points. The dual representation of the EVaR is closely related to the Kullback-Leibler divergence, also known as the relative entropy. Inspired by this dual representation, we define a large class of coherent risk measures, called g-entropic risk measures. The new class includes both the CVaR and the EVaR.

This paper concentrates on solving the portfolio selection problem. It starts with an extension of the well-known optimization framework for Conditional Value-at-Risk (CVaR)-based portfolio selection problems [1, 2] to optimization over a more general class of risk measure - known as the class of Coherent Distortion Risk Measure (CDRM). The CDRM class of risk measures is the intersection of Coherent Risk Measure (CRM) and Distortion Risk Measure (DRM). It concludes with showing that many of the well-known risk measures are of special cases of the CDRM class what may facilitate to deal with the portfolio optimization problem.