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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. [ABSTRACT FROM AUTHOR]

We investigate how the spectral risk measure associated with holding stocks rather than a risk-free deposit, depends on the holding period. Previous papers have shown that within a limited class of spectral risk measures, and when the stock price follows specific processes, spectral risk becomes negative at long periods. We generalize this result for arbitrary exponential Lévy processes. We also prove the same behavior for all spectral risk measures (including the important special case of Expected Shortfall) when the stock price grows realistically fast and when it follows a geometric Brownian motion or a finite moment log stable process. This result would suggest that holding stocks for long periods has a vanishing downside risk. However, using realistic models, we find numerically that spectral risk initially increases for a significant amount of time and reaches zero level only after several decades. Therefore, we conclude that holding stocks has spectral risk for all practically relevant periods. [ABSTRACT FROM AUTHOR]

The intuition of risk is based on two main concepts: loss and variability. In this paper, we present a composition of risk and deviation measures, which contemplate these two concepts. Based on the proposed Limitedness axiom, we prove that this resulting composition, based on properties of the two components, is a coherent risk measure. Similar results for the cases of convex and co-monotone risk measures are exposed. We also provide examples of known and new risk measures constructed under this framework in order to highlight the importance of our approach, especially the role of the Limitedness axiom. [ABSTRACT FROM AUTHOR]

This paper is devoted to the investment strategies that combine asset pricing models and coherent risk measures. In particular, we utilize the theoretical framework of Balbas et al. (J Risk 18(4):25–52, 2016), which suggests that simply by managing a portfolio of assets, an investor can achieve risk that converges to −∞ and returns that converge to + ∞. We contribute on that framework by providing evidence that arise from the CAPM model, in regard to the efficient market hypothesis. In addition, our results suggest that an investor can exhibit returns that outperform the market index by managing a portfolio less volatile than the market. [ABSTRACT FROM AUTHOR]

A classical result in risk measure theory states that every coherent risk measure has a dual representation as the supremum of certain expected value over a risk envelope. We study this topic in more detail. The related issues include: (1) Set operations of risk envelopes and how they change the risk measures, (2) The structure of risk envelopes of popular risk measures, (3) Aversity of risk measures and its impact to risk envelopes, and (4) A connection between risk measures in stochastic optimization and uncertainty sets in robust optimization. [ABSTRACT FROM AUTHOR]

In this work, we study the problem of detecting risk-averse low-diameter clusters in graphs. It is assumed that the clusters represent k-clubs and that uncertain information manifests itself in the form of stochastic vertex weights whose joint distribution is known. The goal is to find a k-club of minimum risk contained in the graph. A stochastic programming framework that is based on the formalism of coherent risk measures is used to quantify the risk of a cluster. We show that the selected representation of risk guarantees that the optimal subgraphs are maximal clusters. A combinatorial branch-and-bound algorithm is proposed and its computational performance is compared with an equivalent mathematical programming approach for instances with k=2,3, and 4. [ABSTRACT FROM AUTHOR]

We study the exploitation of a one species, multiple stand forest plantation when timber price is governed by a stochastic process. Our model is a stochastic dynamic program with a weighted mean-risk objective function, and our main risk measure is the Conditional Value-at-Risk. We consider two stochastic processes, geometric Brownian motion and Ornstein-Uhlenbeck: in the first case, we completely characterize the optimal policy for all possible choices of the parameters while in the second, we provide sufficient conditions assuring that harvesting everything available is optimal. In both cases we solve the problem theoretically for every initial condition. We compare our results with the risk neutral framework and generalize our findings to any coherent risk measure that is affine on the current price. [ABSTRACT FROM AUTHOR]

This paper addresses the Entropic Value-at-Risk ( $${{\mathrm{\mathsf {EV@R}}}}$$ ), a recently introduced coherent risk measure. It is demonstrated that the norms defined by $${{\mathrm{\mathsf {EV@R}}}}$$ induce the same Banach spaces, irrespective of the confidence level. Three vector spaces, called the primal, dual, and bidual entropic spaces, corresponding with $${{\mathrm{\mathsf {EV@R}}}}$$ are fully studied. It is shown that these spaces equipped with the norms induced by $${{\mathrm{\mathsf {EV@R}}}}$$ are Banach spaces. The entropic spaces are then related to the $$L^p$$ spaces, as well as specific Orlicz hearts and Orlicz spaces. This analysis indicates that the primal and bidual entropic spaces can be used as very flexible model spaces, larger than $$L^\infty $$ , over which all $$L^p$$ -based risk measures are well-defined. The dual $${{\mathrm{\mathsf {EV@R}}}}$$ norm and corresponding Hahn-Banach functionals are presented in closed form, which are not explicitly known for the Orlicz and Luxemburg norms that are equivalent to the $${{\mathrm{\mathsf {EV@R}}}}$$ norm. The duality relationships among the entropic spaces are investigated. The duality results are also used to develop an extended Donsker-Varadhan variational formula, and to explicitly provide the dual and Kusuoka representations of $${{\mathrm{\mathsf {EV@R}}}}$$ , as well as the corresponding maximizing densities in both representations. Our results indicate that financial concepts can be successfully used to develop insightful tools for not only the theory of modern risk measurement but also other fields of stochastic analysis and modeling. [ABSTRACT FROM AUTHOR]

Stochastic models used for pricing, reserving, or capital modelling in insurance companies are often very complex, which is why resulting distributions are typically approximated by Monte Carlo simulations. Both the market and regulators exert increasing pressure not to discard the resulting sample distributions, but rather to store them for future review, audit, or validation, as well as to transfer them between actuarial systems. The present work introduces a compression algorithm which approximates an empirical univariate distribution function through a piecewise linear distribution. In contrast to keeping the full sample, such an approximation facilitates the storage and data transfer of the results by drastically reducing memory requirements. The approximation algorithm preserves the mean and imposes a uniformly bounded relative error over a space of coherent risk measures (TVaR). An efficient, open source implementation is provided. [ABSTRACT FROM AUTHOR]

Kusuoka representations provide an important and useful characterization of law invariant coherent risk measures in atomless probability spaces. However, the applicability of these results is limited by the fact that such representations do not always exist in probability spaces with atoms, such as finite probability spaces. We introduce the class of functionally coherent risk measures, which allow us to use Kusuoka representations in any probability space. We show that this class contains every law invariant risk measure that can be coherently extended to a family containing all finite discrete distributions. Thus, it is possible to preserve the desirable properties of law invariant coherent risk measures on atomless spaces without sacrificing generality. We also specialize our results to risk measures on finite probability spaces, and prove a denseness result about the family of risk measures with finite Kusuoka representations. [ABSTRACT FROM AUTHOR]

We extend the classic Mitra and Wan forestry model by assuming that prices follow a geometric Brownian motion. We move one step further in the model with stochastic prices and include risk aversion in the objective function. We prove that, as in the deterministic case, the optimal program is periodic both in the risk neutral and risk averse frameworks, when the benefit function is linear. We find the optimal rotation ages in both stochastic cases and show that they may differ significantly from the deterministic rotation age. In addition, we show how the drift of the price process affects the optimal rotation age and how the degree of risk aversion shortens it. We illustrate our findings for an example of a biomass function and for different values of the model's parameters. [ABSTRACT FROM AUTHOR]

In this paper, we address a basic production planning problem with price dependent demand and stochastic yield of production. We use price and target quantity as decision variables to lower the risk of low yield. The value of risk control becomes more important especially for products with short life cycle. This is because, the profit implications of low yield might be unbearable in the short run. We apply Conditional Value at Risk (CVaR) to model the risk. CVaR measure is a coherent risk measure and thereby having nice conceptual and mathematical underpinnings. It is also widely used in practice. We consider the problem under general demand function and general distribution function of yield and find sufficient conditions under which the problem has a unique local maximum. We also both analytically and numerically analyze the impact of parameter change on the optimal solution. Among our results, we analytically show that with increasing risk aversion, the optimal price increases. This relation is opposite to that of in Newsvendor problem where the uncertainty lies in demand side. [ABSTRACT FROM AUTHOR]

In this work, we consider a class of risk-averse maximum weighted subgraph problems (R-MWSP). Namely, assuming that each vertex of the graph is associated with a stochastic weight, such that the joint distribution is known, the goal is to obtain a subgraph of minimum risk satisfying a given hereditary property. We employ a stochastic programming framework that is based on the formalism of modern theory of risk measures in order to find minimum-risk hereditary structures in graphs with stochastic vertex weights. The introduced form of risk function for measuring the risk of subgraphs ensures that optimal solutions of R-MWS problems represent maximal subgraphs. A graph-based branch-and-bound (BnB) algorithm for solving the proposed problems is developed and illustrated on a special case of risk-averse maximum weighted clique problem. Numerical experiments on randomly generated Erdös-Rényi graphs demonstrate the computational performance of the developed BnB. [ABSTRACT FROM AUTHOR]

In this article, we prove that the minimization problem of the expected shortfall over a convex but not necessarily closed set of financial positions $ \mathcal{X}\subseteq {L^1} $ has a solution. We provide both minimax and variational approaches on this problem. In the case where the optimization conclusions arise from the application of subgradient arguments, we need the assumption that the set of financial positions $ \mathcal{X} $ is closed. [ABSTRACT FROM AUTHOR]

New versions of the set-valued average value at risk for multivariate risks are introduced by generalizing the well-known certainty equivalent representation to the set-valued case. The first 'regulator' version is independent from any market model whereas the second version, called the market extension, takes trading opportunities into account. Essential properties of both versions are proven and an algorithmic approach is provided which admits to compute the values of both versions over finite probability spaces. Several examples illustrate various features of the theoretical constructions. [ABSTRACT FROM AUTHOR]

We examine two important claims by S.S. Wang and J. Treussard concerning the use of distortion functions as a universal tool in pricing financial and insurance risks, and the use of risk neutral probabilities in evaluating risks, respectively. Their claims seem reasonable only in the classical framework of Black-Scholes model, but not convincing in more extended and realistic models such as Lévy processes. [ABSTRACT FROM AUTHOR]

We generalize the notion of arbitrage based on the coherent risk measure, and investigate a mathematical optimization approach for tightening the lower and upper bounds of the price of contingent claims in incomplete markets. Due to the dual representation of coherent risk measures, the lower and upper bounds of price are located by solving a pair of semi-infinite linear optimization problems, which further reduce to linear optimization when conditional value-at-risk (CVaR) is used as risk measure. We also show that the hedging portfolio problem is viewed as a robust optimization problem. Tuning the parameter of the risk measure, we demonstrate by numerical examples that the two bounds approach to each other and converge to a price that is fair in the sense that seller and buyer face the same amount of risk. [ABSTRACT FROM AUTHOR]

Set-valued risk measures on $${L^p_d}$$ with 0 ≤ p ≤ ∞ for conical market models are defined, primal and dual representation results are given. The collection of initial endowments which allow to super-hedge a multivariate claim are shown to form the values of a set-valued sublinear (coherent) risk measure. Scalar risk measures with multiple eligible assets also turn out to be a special case within the set-valued framework. [ABSTRACT FROM AUTHOR]

This work introduces a new analytical approach to the formulation of optimization problems with piecewise-defined (PD) objective functions. First, we introduce a new definition of multivariate PD functions and derive formal results for their continuity and differentiability. Then, we obtain closed-form expressions for the calculation of their moments. We apply these findings to three classes of optimization problems involving coherent risk measures. The method enables one to obtain insights on problem structure and on sensitivity to imprecision at the problem formulation stage, eliminating reliance on ad-hoc post-optimality numerical calculations. [ABSTRACT FROM AUTHOR]

Spectral risk measures (SRMs) are risk measures that take account of user risk-aversion, but to date there has been little guidance on the choice of utility function underlying them. This paper addresses this issue by examining alternative approaches based on exponential and power utility functions. A number of problems are identified with both types of spectral risk measure. The general lesson is that users of spectral risk measures must be careful to select utility functions that fit the features of the particular problems they are dealing with, and should be especially careful when using power SRMs. [ABSTRACT FROM AUTHOR]

In this paper we discuss computational complexity and risk averse approaches to two and multistage stochastic programming problems. We argue that two stage (say linear) stochastic programming problems can be solved with a reasonable accuracy by Monte Carlo sampling techniques while there are indications that complexity of multistage programs grows fast with increase of the number of stages. We discuss an extension of coherent risk measures to a multistage setting and, in particular, dynamic programming equations for such problems. [ABSTRACT FROM AUTHOR]

We give an explicit PDE characterization for the solution of the problem of maximizing the utility of both terminal wealth and intertemporal consumption under model uncertainty. The underlying market model consists of a risky asset, whose volatility and long-term trend are driven by an external stochastic factor process. The robust utility functional is defined in terms of a HARA utility function with risk aversion parameter 0 < α < 1 and a dynamically consistent coherent risk measure, which allows for model uncertainty in the distributions of both the asset price dynamics and the factor process. Ourmethod combines recent results by Wittmüß (Robust optimization of consumption with random endowment, 2006) on the duality theory of robust optimization of consumption with a stochastic control approach to the dual problem of determining a ‘worst-case martingale measure’. [ABSTRACT FROM AUTHOR]

We prove that on an atomless probability space, every dilatation monotone convex risk measure is law invariant. This result, combined with the known ones, shows the equivalence between dilatation monotonicity and important properties of convex risk measures such as law invariance and second-order stochastic monotonicity. [ABSTRACT FROM AUTHOR]

Assume that the random future evolution of values is modelled in continuous time. Then, a risk measure can be viewed as a functional on a space of continuous-time stochastic processes. In this paper we study coherent and convex monetary risk measures on the space of all càdlàg processes that are adapted to a given filtration. We show that if such risk measures are required to be real-valued, then they can only depend on a stochastic process in a way that is uninteresting for many applications. Therefore, we allow them to take values in ( −∞, ∞]. The economic interpretation of a value of ∞ is that the corresponding financial position is so risky that no additional amount of money can make it acceptable. The main result of the paper gives different characterizations of coherent or convex monetary risk measures on the space of all bounded adapted càdlàg processes that can be extended to coherent or convex monetary risk measures on the space of all adapted càdlàg processes. As examples we discuss a new approach to measure the risk of an insurance company and a coherent risk measure for unbounded càdlàg processes induced by a so called m-stable set. [ABSTRACT FROM AUTHOR]

The paper deals with the study of a coherent risk measure, which we call Weighted V@R. It is a risk measure of the form $$\rho_\mu(X)=\int\limits_{[0,1]}\hbox{TV}@\hbox{R}_{\uplambda}(\hbox{X}) \mu(\hbox{d}\uplambda),$$ where μ is a probability measure on [0,1] and TV@R stands for Tail V@R. After investigating some basic properties of this risk measure, we apply the obtained results to the financial problems of pricing, optimization, and capital allocation. It turns out that, under some regularity conditions on μ, Weighted V@R possesses some nice properties that are not shared by Tail V@R. To put it briefly, Weighted V@R is “smoother” than Tail V@R. This allows one to say that Weighted V@R is one of the most important classes (or maybe the most important class) of coherent risk measures. [ABSTRACT FROM AUTHOR]

We consider a regime-switching HJB approach to evaluate risk measures for derivative securities when the price process of the underlying risky asset is governed by the exponential of a pure jump process with drift and a Markov switching compensator. The pure jump process is flexible enough to incorporate both the infinite, (small), jump activity and the finite, (large), jump activity. The drift and the compensator of the pure jump process switch over time according to the state of a continuous-time hidden Markov chain representing the state of an economy. The market described by our model is incomplete. Hence, there is more than one pricing kernel and there is no perfect hedging strategy for a derivative security. We derive the regime-switching HJB equations for coherent risk measures for the unhedged position of derivative securities, including standard European options and barrier options. For measuring risk inherent in the unhedged option position, we first need to mark the position into the market by valuing the option. We employ a well-known tool in actuarial science, namely, the Esscher transform to select a pricing kernel for valuation of an option and to generate a family of real-world probabilities for risk measurement. We also derive the regime-switching HJB-variational inequalities for coherent risk measures for American-style options. [ABSTRACT FROM AUTHOR]

Risk management by applying operational flexibility is becoming a key issue for production companies. This paper discusses how a power portfolio can be hedged through its own production assets. In particular we model operational flexibility of a hydro pump storage plant and show how to dispatch it to hedge against adverse movements in the portfolio. Moreover, we present how volume risk, which is not hedgeable with standard contracts from power exchanges, can be managed by an intelligent dispatch policy. Despite the incompleteness of the market we quantify the value of this operational flexibility in the framework of coherent risk measures. [ABSTRACT FROM AUTHOR]

General deviation measures are introduced and studied systematically for their potential applications to risk management in areas like portfolio optimization and engineering. Such measures include standard deviation as a special case but need not be symmetric with respect to ups and downs. Their properties are explored with a mind to generating a large assortment of examples and assessing which may exhibit superior behavior. Connections are shown with coherent risk measures in the sense of Artzner, Delbaen, Eber and Heath, when those are applied to the difference between a random variable and its expectation, instead of to the random variable itself. However, the correspondence is only one-to-one when both classes are restricted by properties called lower range dominance, on the one hand, and strict expectation boundedness on the other. Dual characterizations in terms of sets called risk envelopes are fully provided. [ABSTRACT FROM AUTHOR]

Assume that the random future evolution of values is modelled in continuous time. Then, a risk measure can be viewed as a functional on a space of continuous-time stochastic processes. In this paper we study coherent and convex monetary risk measures on the space of all cádlág processes that are adapted to a given filtration. We show that if such risk measures are required to be real-valued, then they can only depend on a stochastic process in a way that is uninteresting for many applications. Therefore, we allow them to take values in. The economic interpretation of a value ofis that the corresponding financial position is so risky that no additional amount of money can make it acceptable. The main result of the paper gives different characterizations of coherent or convex monetary risk measures on the space of all bounded adapted cádlág processes that can be extended to coherent or convex monetary risk measures on the space of all adapted cádlág processes. As examples we discuss a new approach to measure the risk of an insurance company and a coherent risk measure for unbounded cádlág processes induced by a so called m-stable set. [ABSTRACT FROM AUTHOR]

We define (d,n)-coherent risk measures as set-valued maps fromintosatisfying some axioms. We show that this definition is a convenient extension of the real-valued risk measures introduced by Artzner et al. [2]. We then discuss the aggregation issue, i.e., the passage fromvalued random portfolio tovalued measure of risk. Necessary and sufficient conditions of coherent aggregation are provided. [ABSTRACT FROM AUTHOR]