Coherent Risk Measures Research Articles

Intra‐Horizon expected shortfall and risk structure in models with jumps

AbstractThe present article deals with intra‐horizon risk in models with jumps. Our general understanding of intra‐horizon risk is along the lines of the approach taken in Boudoukh et al. (2004); Rossello (2008); Bhattacharyya et al. (2009); Bakshi and Panayotov (2010); and Leippold and Vasiljević (2020). In particular, we believe that quantifying market risk by strictly relying on point‐in‐time measures cannot be deemed a satisfactory approach in general. Instead, we argue that complementing this approach by studying measures of risk that capture the magnitude of losses potentially incurred at any time of a trading horizon is necessary when dealing with (m)any financial position(s). To address this issue, we propose an intra‐horizon analogue of the expected shortfall for general profit and loss processes and discuss its key properties. Our intra‐horizon expected shortfall is well‐defined for (m)any popular class(es) of Lévy processes encountered when modeling market dynamics and constitutes a coherent measure of risk, as introduced in Cheridito et al. (2004). On the computational side, we provide a simple method to derive the intra‐horizon risk inherent to popular Lévy dynamics. Our general technique relies on results for maturity‐randomized first‐passage probabilities and allows for a derivation of diffusion and single jump risk contributions. These theoretical results are complemented with an empirical analysis, where popular Lévy dynamics are calibrated to the S&P 500 index and Brent crude oil data, and an analysis of the resulting intra‐horizon risk is presented.

A Framework for Power System Operational Planning Under Uncertainty Using Coherent Risk Measures

With the increasing integration of renewable energy sources (RESs) and the implementation of dynamic line rating (DLR), the accompanying uncertainties in power systems require intensive management to ensure reliable and secure operational planning. However, while numerous approaches and methods in the literature deal with uncertainties, they have not been analyzed axiomatically. This paper presents an analysis of risk in power system operation using coherent risk measures, elaborating on the origin of risk and the mechanisms of its management in the presence of various sources of uncertainty. To illustrate the practicality and benefits of coherent risk measures, a risk-averse asymmetry robust unit commitment (UC) model is established. It is based on coherent reformulations of the uncertain reserve and line flow constraints and is formulated in the form of a compact computationally efficient mixed-integer second-order conic program (SOCP). The overall performance of the proposed framework is verified using the updated 2019 IEEE Reliability Test System and the ACTIVSg2000 test system over a year-long period.

A primal–dual algorithm for risk minimization

In this paper, we develop an algorithm to efficiently solve risk-averse optimization problems posed in reflexive Banach space. Such problems often arise in many practical applications as, e.g., optimization problems constrained by partial differential equations with uncertain inputs. Unfortunately, for many popular risk models including the coherent risk measures, the resulting risk-averse objective function is nonsmooth. This lack of differentiability complicates the numerical approximation of the objective function as well as the numerical solution of the optimization problem. To address these challenges, we propose a primal–dual algorithm for solving large-scale nonsmooth risk-averse optimization problems. This algorithm is motivated by the classical method of multipliers and by epigraphical regularization of risk measures. As a result, the algorithm solves a sequence of smooth optimization problems using derivative-based methods. We prove convergence of the algorithm even when the subproblems are solved inexactly and conclude with numerical examples demonstrating the efficiency of our method.

Pareto efficient buy and hold investment strategies under order book linked constraints

This paper deals with a multiobjective portfolio selection problem involving expected wealth (or return), coherent risk measures, deviation measures, and “level II order book data”, i.e., natural restrictions provoked by the existence of several levels of both bid and ask quotes with the corresponding depth. Obviously, the incorporation of the order book information makes our study much more realistic, since it is more closely related to the real behavior of many financial markets. Besides, ambiguity may be incorporated, which allows us to overcome the model-risk, since a potential discrepancy between the real (and maybe unknown) probabilities and the estimated ones is taken into account. Though the portfolio choice problem is not at all linear, its dual problem becomes a linear goal programming problem, and, consequently, the absence of duality gap allows us to solve the portfolio choice problem in an easy way. Furthermore, the double dual problem is linear too, and its solution also leads to the Pareto-efficient frontier. Lastly, we explore the influence on the Pareto-efficient frontier of the existence of “golden strategies”, i.e., investment strategies whose tail (or downside) risk is strictly lower than their price. Numerical experiments involving all the findings are implemented in a derivative market where both future contracts and call/put options may be traded.

A Theory for Measures of Tail Risk

The notion of “tail risk” has been a crucial consideration in modern risk management and financial regulation, as very well documented in the recent regulatory documents. To achieve a comprehensive understanding of the tail risk, we carry out an axiomatic study for risk measures that quantify the tail risk, that is, the behaviour of a risk beyond a certain quantile. Such risk measures are referred to as tail risk measures in this paper. The two popular classes of regulatory risk measures in banking and insurance, value at risk (VaR) and expected shortfall, are prominent, yet elementary, examples of tail risk measures. We establish a connection between a tail risk measure and a corresponding law-invariant risk measure, called its generator, and investigate their joint properties. A tail risk measure inherits many properties from its generator, but not subadditivity or convexity; nevertheless, a tail risk measure is coherent if and only if its generator is coherent. We explore further relevant issues on tail risk measures, such as bounds, distortion risk measures, risk aggregation, elicitability, and dual representations. In particular, there is no elicitable tail convex risk measure other than the essential supremum, and under a continuity condition, the only elicitable and positively homogeneous monetary tail risk measures are the VaRs.

Due-date quotation model for manufacturing system scheduling under uncertainty

This paper studies the scheduling problem for the manufacturing systems with uncertain job duration, and the possibility of planning due-date quotations for critical manufacturing tasks given a fixed contingency budget. We propose a due-date quotation model to measure the risk of delay in the manufacturing process in terms of the allocated contingency budget. The risk of delay is measured in the same unit as its corresponding milestone factor such that the decision makers could directly visualize and quantify the level of risks in units of hours or days. In addition, the proposed model possesses various great properties required by a convex risk measure and it represents a minimized certainty equivalent of the overall expected risk in achieving the manufacturing due-dates. Extensive computational experiments are conducted to evaluate the model performance. The results show that our proposed model, compared to various existing methods, provides a much more balanced performance in terms of success rate of due-date achievement, due-date quotation shortfall, as well as, robustness against uncertainties. The practical applicability of the proposed models are also tested with the job scheduling problem in a real stamping industry application.

Actuarial Pricing with Nancial Methods

The objective of this paper is twofold. On the one hand, the optimal combination of reinsurance and financial investment is studied under a general framework, since there is no specific type of reinsurance contract, there is no specific dynamics of the financial instruments, the financial market does not have to be free of frictions and there are no constraints with respect to the financial securities to be selected, that is, derivative assets may be involved too. On the other hand, it is pointed out how the optimal combination above may provide us with new premium principles making the global insurer risk vanish, this risk being computed with a co- coherent risk measure. The new premium principles seem to respect several properties which are desirable from both the analytical and the economic perspectives. Indeed, from an analytical viewpoint, they are continuous, homogeneous, and increasing, while from the economic viewpoint the premium principles lead to cheaper prices with respect to both the insurance market and the financial market. In other words, the premium principles make the insurer more competitive though the insurer's global risk is vanishing. General necessary and sufficient optimality conditions will be given, as well as closed forms for the solutions under appropriate assumptions. Special attention will deserve those methods preventing unbounded optimization problems and a particular case will be more thoroughly studied, namely, the combination of the Black-Scholes- Merton pricing model with the conditional value at risk.

An Interior-Point Approach for Solving Risk-Averse PDE-Constrained Optimization Problems with Coherent Risk Measures

The prevalence of uncertainty in models of engineering and the natural sciences necessitates the inclusion of random parameters in the underlying partial differential equations (PDEs). The resulting decision problems governed by the solution of such random PDEs are infinite dimensional stochastic optimization problems. In order to obtain risk-averse optimal decisions in the face of such uncertainty, it is common to employ risk measures in the objective function. This leads to risk-averse PDE-constrained optimization problems. We propose a method for solving such problems in which the risk measures are convex combinations of the mean and conditional value-at-risk (CVaR). Since these risk measures can be evaluated by solving a related inequality-constrained optimization problem, we suggest a log-barrier technique to approximate the risk measure. This leads to a new continuously differentiable convex risk measure: the log-barrier risk measure. We show that the log-barrier risk measure fits into the setting of optimized certainty equivalents of Ben-Tal and Teboulle and the expectation quadrangle of Rockafellar and Uryasev. Using the differentiability of the log-barrier risk measure, we derive first-order optimality conditions reminiscent of classical primal and primal-dual interior-point approaches in nonlinear programming. We derive the associated Newton system, propose a reduced symmetric system to calculate the steps, and provide a sufficient condition for local superlinear convergence in the continuous setting. Furthermore, we provide a $\Gamma$-convergence result for the log-barrier risk measures to prove convergence of the minimizers to the original nonsmooth problem. The results are illustrated by a numerical study.

Gini Shortfall: A Gini mean difference-based risk measure

A risk is the possibility of undesirable events happening in the future. A good risk measure is needed to quantify the risks one faces. Some well-known risk measures include the Value-at-Risk (VaR) and the Expected Shortfall (ES). VaR measures the lower bound for big losses in a loss distribution tail, while ES measures the average of losses surpassing the VaR. Unfortunately, there are drawbacks in using the stated risk measures, mainly that they do not provide information regarding the variability of losses in the distribution tail. This paper will introduce and explore Gini Shortfall (GS), a more comprehensive risk measure than VaR and ES. GS provides information on the variability of data in distribution tails measured with Tail-Gini functional, a tail variability measure based on the variability measure Gini Mean Difference or Gini functional. Another superiority of GS is that under certain conditions, it can satisfy the four criteria of coherency. A coherent risk measure is useful for companies or investors to determine the right business and investing strategies. This paper will also provide explicit formulas of GS for some loss-related distributions, namely the exponential, Pareto, and logistic distributions. These formulas are then applied to calculate risks from actual stock data.

Kalman--Bucy Filtering and Minimum Mean Square Estimator under Uncertainty

In this paper, we study a generalized Kalman--Bucy filtering problem under uncertainty. The drift uncertainty for both signal process and observation process is considered, and the attitude to uncertainty is characterized by a convex operator (convex risk measure). The optimal filter or the minimum mean square estimator (MMSE) is calculated by solving the minimum mean square estimation problem under a convex operator. In the first part of this paper, this estimation problem is studied under $g$-expectation which is a special convex operator. For this case, we prove that there exists a worst-case prior $P^{\theta^{\ast}}$. Based on this $P^{\theta^{\ast}}$ we obtained the Kalman--Bucy filtering equation under $g$-expectation. In the second part of this paper, we study the minimum mean square estimation problem under general convex operators. The existence and uniqueness results of the MMSE are deduced.

The hedging-based utility risk measure

FinTech makes numerous financial products accessible to common investors but up to now, there is no risk measure method specially customized for common investors instead of financial institutions which are generally too big to fail. This paper develops a hedging-based utility risk measure (HBU) theory. We show that HBU is a convex risk measure and if the utility has a constant relative risk aversion coefficient, HBU is also coherent. Roughly speaking, HBU is the opposite of a generalized utility indifference price. HBU depends on claimants' utility and the hedging instruments accessible to them and thus it is a personal (subjective) risk measure especially suitable for common investors who need to have a comprehensive risk evaluation of a financial product.

A Framework for Measures of Risk under Uncertainty

A risk analyst assesses potential financial losses based on multiple sources of information. Often, the assessment does not only depend on the specification of the loss random variable, but also various economic scenarios. Motivated by this observation, we design a unified axiomatic framework for risk evaluation principles which quantifies jointly a loss random variable and a set of plausible probabilities. We call such an evaluation principle a generalized risk measure. We present a series of relevant theoretical results. The worst-case, coherent, and robust generalized risk measures are characterized via different sets of intuitive axioms. We establish the equivalence between a few natural forms of law invariance in our framework, and the technical subtlety therein reveals a sharp contrast between our framework and the traditional one. Moreover, coherence and strong law invariance are derived from a combination of other conditions, which provides additional support for coherent risk measures such as Expected Shortfall over Value-at-Risk, a relevant issue for risk management practice.

Conditional coherent risk measures and regime-switching conic pricing

<p style='text-indent:20px;'>This paper introduces and represents conditional coherent risk measures as essential suprema of conditional expectations over a convex set of probability measures and as distorted expectations given a concave distortion function. A model is then developed for the bid and ask prices of a European-type asset by a conic formulation. The price process is governed by a modified geometric Brownian motion whose drift and diffusion coefficients depend on a Markov chain. The bid and ask prices of a European-type asset are then characterized using conic quantization.</p>

Estimation of Conditional Weighted Expected Shortfall under Adjusted Extreme Quantile Autoregression

In this paper, we present an estimator that improves the well-calibrated coherent risk measure: expected shortfall by restructuring its functional form to incorporate dynamic weights on extreme conditional quantiles used in its definition. Adjusted Extreme Quantile Autoregression will is used in estimating intermediary location measures. Consistency and coherence of the estimator are also proved. The resulting estimator was found to be less conservative compared to the expected shortfall.

A numerical approach to the risk capital allocation problem

Capital risk allocation is an important problem in corporate, financial and insurance risk management. There are two theoretical aspects to this problem. The first aspect consists of choosing a risk measure, and the second consists of choosing a capital allocation rule subordinated to the risk measure. When these two complementary aspects are settled, the problem is reduced to a computational problem. No universal prescription for choosing a risk measure exists, let alone a subordinated capital allocation rule. However, expert opinion may be expressed as a range of capital to be allocated to each risk. For example, the managers of each line of activity of a corporation or bank may have a way to estimate a minimum and maximum losses with high confidence. In that case, one is confronted with two interesting numerical problems. First, to devise a method to allocate risk capital within the bounds provided by the expert. Second, to use the upper bounds as stand-alone risk prices to determine a risk measure, which can then be used to compute the prices of other risks. The risk measure could also be used to compute a range for the capital allocations of a new collection of risks, to which our numerical procedure to allocate risk capital can be applied. The aim of this paper is to use a model-free, nonparametric approach based on the method of maximum entropy in the mean to solve both problems and to examine how they are intertwined. In this way we circumvent the problem of choosing between risk valuation and risk pricing methods, and between risk capital allocation rules consistent with them. Also, the maximum-entropy-based methodology provides us with a method to determine coherent risk measures consistent with the bounds provided by expert opinion or management. The risk measure provides us with a method to determine either risk prices or bounds to be used for capital allocation for a different collection of risks.

Pro Rata Credit Risk Pooling Between Regional Banks: Economic Capital and Portfolio Granularity

German savings and cooperative banks use credit risk pooling transactions as a specific type of synthetic credit risk transfer. This paper describes the effect of pro rata credit risk pooling transactions on the granularity of these banks’ credit portfolios. The change in granularity is described analytically using general properties of concentration measures. This allows to prove independent of specific concentration measures or credit portfolio models that the granularity of a credit portfolio will increase if a bank participates in pro rata credit risk pooling with homogeneous credit risks. But simulations show that Value-at-Risk will not always decrease if a portfolio’s granularity increases, ceteris paribus. As a consequence, banks using Value-at-Risk instead of a coherent risk measure like Expected Shortfall might choose to limit their participation in credit risk pooling transactions.

Sensitivity to large losses and ρ-arbitrage for convex risk measures

Sensitivity to large losses and ρ-arbitrage for convex risk measures

A Cyber-Secured Operation for Water-Energy Nexus

The wide implementation of information and communication technologies (ICT) cause power system operations exposed to cyber-attacks. Meanwhile, the tendency of integrated multi energy vectors has worsened this issue with multiple energy coupled. This paper proposes a two-stage risk-averse mitigation strategy for water-energy systems (WESs), incorporating power, natural gas and water systems against false data injection attacks (FDIA) under water-energy nexus. The FDIA on individual sub-systems is modelled through hampering false data integrity to the systems. An innovative two-stage risk-averse distributionally robust optimization (RA-DRO) is proposed to mitigate uneconomic operation and provides a coordinated optimal load shedding scheme for the nexus system security. A coherent risk measure, Conditional Value-at-Risk is incorporated into the RA-DRO to model risk. A Benders decomposition method is used to solve the original NP-hard RA-DRO problem. Case studies are demonstrated on a WES under water-energy nexus and results show that the effectiveness of the method to mitigate risks from potential FDIA and renewable uncertainties. This research provides WES operators an economic system operation tool by optimally coordinating energy infrastructures and implementing reasonable load shedding to enhance cybersecurity.

Nonparametric extreme conditional expectile estimation

AbstractExpectiles and quantiles can both be defined as the solution of minimization problems. Contrary to quantiles though, expectiles are determined by tail expectations rather than tail probabilities, and define a coherent risk measure. For these two reasons in particular, expectiles have recently started to be considered as serious candidates to become standard tools in actuarial and financial risk management. However, expectiles and their sample versions do not benefit from a simple explicit form, making their analysis significantly harder than that of quantiles and order statistics. This difficulty is compounded when one wishes to integrate auxiliary information about the phenomenon of interest through a finite‐dimensional covariate, in which case the problem becomes the estimation of conditional expectiles. In this paper, we exploit the fact that the expectiles of a distribution F are in fact the quantiles of another distribution E explicitly linked to F, in order to construct nonparametric kernel estimators of extreme conditional expectiles. We analyze the asymptotic properties of our estimators in the context of conditional heavy‐tailed distributions. Applications to simulated data and real insurance data are provided.

Rules for a coherent real estate risk scoring

Purpose Scoring is a widely used, long-established, and universally applicable method of measuring risks, especially those that are difficult to quantify. Unfortunately, the scoring method is often misused in real estate practice and underestimated in academia. The purpose of this paper is to supplement the literature with general rules under which scoring systems should be designed and validated, so that they can become reliable risk instruments. Design/methodology/approach The paper combines the rules, or axioms, for coherent risk measures known from the literature with those for scoring instruments. The result is a system of rules that a risk scoring system should fulfil. The approach is theoretical, based on a literature survey and reasoning. Findings At first, the paper clarifies that a risk score should express the variation of a property’s yield and not of its quality, as it is often done in practice. Then the axioms for a coherent risk scoring are derived, e.g. the independence of the risk factors. Finally, the paper proposes procedures for valid and reliable risk scoring systems, e.g. the out-of-time validation. Practical implications Although it is a theoretical work, the paper also focuses on practical applicability. The findings are illustrated with examples of scoring systems. Originality/value Rules for risk measures and for scoring systems have been established long ago, but the combination is a first. In this way, the paper contributes to real estate risk research and risk management practice.