Class Of Coherent Risk Measures Research Articles

Robust Portfolio Optimization with Respect to Spectral Risk Measures Under Correlation Uncertainty

This paper proposes a distributionally robust multi-period portfolio model with ambiguity on asset correlations with fixed individual asset return mean and variance. The correlation matrix bounds can be quantified via corresponding confidence intervals based on historical data. We employ a general class of coherent risk measures namely the spectral risk measure, which includes the popular measure conditional value-at-risk (CVaR) as a particular case, as our objective function. Specific choices of spectral risk measure permit flexibility for capturing risk preferences of different investors. A semi-analytical solution is derived for our model. The prominent stochastic dual dynamic programming (SDDP) algorithm adapted with intricate modifications is developed as a numerical method under the discrete distribution setting. In particular, our new formulation accounts for the unknown worst-case distribution in each iteration. We verify the convergence property of this algorithm under the setting of finite scenarios. Our results show that the optimal solution favours a certain degree of anti-diversification due to dependence ambiguity and exhibits its protection ability during the financial crisis period.

On Coherent Risk Measures Induced by Convex Risk Measures

We study the close relationship between coherent risk measures and convex risk measures. Inspired by the obtained results, we propose a class of coherent risk measures induced by convex risk measures. The robust representation and minimization problem of the induced coherent risk measure are investigated. A new coherent risk measure, the Entropic Conditional Value-at-Risk (ECVaR), is proposed as a special case. We show how to apply the induced coherent risk measure to realistic portfolio selection problems. Finally, by comparing its out-of-sample performance with that of CVaR, entropic risk measure, as well as entropic value-at-risk, we carry out a series of empirical tests to demonstrate the practicality and superiority of the ECVaR measure in optimal portfolio selection.

Sequential Decision Making With Coherent Risk

We provide sampling-based algorithms for optimization under a coherent-risk objective. The class of coherent-risk measures is widely accepted in finance and operations research, among other fields, and encompasses popular risk-measures such as conditional value at risk and mean-semi-deviation. Our approach is suitable for problems in which tuneable parameters control the distribution of the cost, such as in reinforcement learning or approximate dynamic programming with a parameterized policy. Such problems cannot be solved using previous approaches. We consider both static risk measures and time-consistent dynamic risk measures. For static risk measures, our approach is in the spirit of policy gradient methods, while for the dynamic risk measures, we use actor-critic type algorithms.

Coherent Risk Measures and Convex Combinations of the Conditional Value at Risk (C V@R)

The conditional-value-at-risk (C V@R) has been widely used as a risk measure. It is well known, that C V@R is coherent in the sense of Artzner, Delbaen, Eber, Heath (1999). The class of coherent risk measures is convex. It was conjectured, that all coherent risk measures can be represented as convex combinations of C V@R’s. In this note we show that this conjecture is wrong.

Coherent risk measures by pricing functionals

The aim of this paper is to introduce a new class of Coherent Risk Measures in Banach lattices, whose dual representation is consistent to the pricing functionals of an incomplete market. As a sub-class, we introduce the corresponding Expected Shortfall Risk Measures.

Optimization with Multivariate Conditional Value-at-Risk Constraints

For many decision-making problems under uncertainty, it is crucial to develop risk-averse models and specify the decision makers' risk preferences based on multiple stochastic performance measures (or criteria). Incorporating such multivariate preference rules into optimization models is a fairly recent research area. Existing studies focus on extending univariate stochastic dominance rules to the multivariate case. However, enforcing multivariate stochastic dominance constraints can often be overly conservative in practice. As an alternative, we focus on the widely applied risk measure conditional value-at-risk (CVaR), introduce a multivariate CVaR relation, and develop a novel optimization model with multivariate CVaR constraints based on polyhedral scalarization. To solve such problems for finite probability spaces, we develop a cut generation algorithm, where each cut is obtained by solving a mixed-integer problem. We show that a multivariate CVaR constraint reduces to finitely many univariate CVaR constraints, which proves the finite convergence of our algorithm. We also show that our results can be naturally extended to a wider class of coherent risk measures. The proposed approach provides a flexible and computationally tractable way of modeling preferences in stochastic multicriteria decision making. We conduct a computational study for a budget allocation problem to illustrate the effect of enforcing multivariate CVaR constraints and demonstrate the computational performance of the proposed solution methods.

In This Issue

In This Issue

A Generalization of the Aumann-Shapley Value for Risk Capital Allocation Problems

This paper analyzes risk capital allocation problems. For risk capital allocation problems, the aim is to allocate the risk capital of a firm to its divisions. Risk capital allocation is of central importance in risk-based performance measurement. We consider a case in which the aggregate risk capital is determined via a coherent risk measure. The academic literature advocates an allocation rule that, in game-theoretic terms, is equivalent to using the Aumann-Shapley value as solution concept. This value is however not well-defined in case a differentiability condition is not satisfied. As an alternative, we introduce an allocation rule inspired by the Shapley value in a fuzzy setting. We take a grid on a fuzzy participation set, define paths on this grid and construct an allocation rule based on a path. Then, we define a rule as the limit of the average over these allocations, when the grid size converges to zero. We introduce this rule for a broad class of coherent risk measures. We show that if the Aumann-Shapley value is well-defined, the allocation rule coincides with it. If the Aumann-Shapley value is not defined, which is due to non-differentiability problems, the allocation rule specifies an explicit allocation. It corresponds with the Mertens value, which is originally characterized in an axiomatic way (Mertens, 1988), whereas we provide an asymptotic argument.

Entropic Value-at-Risk: A New Coherent Risk Measure

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.

Constructing Uncertainty Sets for Robust Linear Optimization

In this paper, we propose a methodology for constructing uncertainty sets within the framework of robust optimization for linear optimization problems with uncertain parameters. Our approach relies on decision maker risk preferences. Specifically, we utilize the theory of coherent risk measures initiated by Artzner et al. (1999) [Artzner, P., F. Delbaen, J. Eber, D. Heath. 1999. Coherent measures of risk. Math. Finance 9 203–228.], and show that such risk measures, in conjunction with the support of the uncertain parameters, are equivalent to explicit uncertainty sets for robust optimization. We explore the structure of these sets in detail. In particular, we study a class of coherent risk measures, called distortion risk measures, which give rise to polyhedral uncertainty sets of a special structure that is tractable in the context of robust optimization. In the case of discrete distributions with rational probabilities, which is useful in practical settings when we are sampling from data, we show that the class of all distortion risk measures (and their corresponding polyhedral sets) are generated by a finite number of conditional value-at-risk (CVaR) measures. A subclass of the distortion risk measures corresponds to polyhedral uncertainty sets symmetric through the sample mean. We show that this subclass is also finitely generated and can be used to find inner approximations to arbitrary, polyhedral uncertainty sets.

A new class of coherent risk measures based on p-norms and their applications

To exercise better control on the lower tail of the loss distribution and to easily describe the investor's risk attitude, a new class of coherent risk measures is proposed in this paper by taking ...

A new class of coherent risk measures based on p‐norms and their applications

AbstractTo exercise better control on the lower tail of the loss distribution and to easily describe the investor's risk attitude, a new class of coherent risk measures is proposed in this paper by taking the minimization of p‐norms of lower losses with respect to some reference point. We demonstrate that the new risk measure has satisfactory mathematical properties such as convexity, continuity with respect to parameters included in its definition, the relations between two new risk measures are also examined. The application of the new risk measures for optimal portfolio selection is illustrated by using trade data from the Chinese stock markets. Empirical results not only support our theoretical conclusions, but also show the practicability of the portfolio selection model with our new risk measures. Copyright © 2006 John Wiley & Sons, Ltd.

Dynamic coherent risk measures

Monetary measures of risk like Value at Risk or Worst Conditional Expectation assess the risk of financial positions. The existing risk measures are of a static, one period nature. In this paper, I define dynamic monetary risk measures and I present an axiomatic approach that extends the class of coherent risk measures to the dynamic framework. The axiom of translation invariance has to be recast as predictable translation invariance to account for the release of new information. In addition to the coherency axioms, I introduce the axiom of dynamic consistency. Consistency requires that judgements based on the risk measure are not contradictory over time. I show that consistent dynamic coherent risk measures can be represented as the worst conditional expectation of discounted future losses where the expectations are being taken over a set of probability measures that satisfies a consistency condition.

The optimal portfolio problem with coherent risk measure constraints

One of the basic problems of applied finance is the optimal selection of stocks, with the aim of maximizing future returns and constraining risks by an appropriate measure. Here, the problem is formulated by finding the portfolio that maximizes the expected return, with risks constrained by the worst conditional expectation. This model is a straightforward extension of the classic Markovitz mean–variance approach, where the original risk measure, variance, is replaced by the worst conditional expectation. The worst conditional expectation with a threshold α of a risk X, in brief WCE α ( X), is a function that belongs to the class of coherent risk measures. These are measures that satisfy a set of properties, such as subadditivity and monotonicity, that are introduced to prevent some of the drawbacks that affect some other common measures. This paper shows that the optimal portfolio selection problem can be formulated as a linear programming instance, but with an exponential number of constraints. It can be solved efficiently by an appropriate generation constraint subroutine, so that only a small number of inequalities are actually needed. This method is applied to the optimal selection of stocks in the Italian financial market and some computational results suggest that the optimal portfolios are better than the market index.

Risk capital allocation by coherent risk measures based on one-sided moments

This paper proposes differentiability properties for positively homogeneous risk measures which ensure that the gradient can be applied for reasonable risk capital allocation on non-trivial portfolios. It is shown that these properties are fulfilled for a wide class of coherent risk measures based on the mean and the one-sided moments of a risky payoff. In contrast to quantile-based risk measures like Value-at-Risk (VaR), this class allows allocation in portfolios of very general distributions, e.g. discrete ones. Two examples show how risk capital given by the VaR can be allocated by adapting risk measures of this class to the VaR.

Coherent risk measurement: an introduction

It is shown that the widely used risk measures standard deviation and value at risk do not always reflect risk preferences accurately. To overcome these problems in risk measurement a class of coherent risk measures has been proposed. We introduce the idea behind these measures and provide an overview of suggested coherent risk measures. Finally it is shown where the limitations of such measures in practical applications are and how regulatory bodies responded to their introduction in the literature. We find that most contributions on coherent risk measurement come from the actuarial sciences and propagate a widening of the discussion among researchers and practitioners in other industries.

Expected shortfall and beyond

Financial institutions have to allocate so-called economic capital in order to guarantee solvency to their clients and counterparties. Mathematically speaking, any methodology of allocating capital is a risk measure, i.e. a function mapping random variables to the real numbers. Nowadays value-at-risk (VaR), which is defined as a fixed level quantile of the random variable under consideration, is the most popular risk measure. Unfortunately, it fails to reward diversification, as it is not subadditive. In the search for a suitable alternative to VaR, expected shortfall (ES) (or conditional VaR or tail VaR) has been characterized as the smallest coherent and law invariant risk measure to dominate VaR. We discuss these and some other properties of ES as well as its generalization to a class of coherent risk measures which can incorporate higher moment effects. Moreover, we suggest a general method on how to attribute ES risk contributions to portfolio components.