Multivariate risks and depth-trimmed regions

We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Risk, while the zonoid trimming yields the expected shortfall. In the abstract framework, it is shown how to establish a both-ways correspondence between risk measures and depth-trimmed regions. It is also demonstrated how the lattice structure of the space of risk values influences this relationship. I. Molchanov supported by Swiss National Science Foundation Grant 200020-109217.     

Dynamic risk measures: Time consistency and risk measures from BMO martingales

Time consistency is a crucial property for dynamic risk measures. Making use of the dual representation for conditional risk measures, we characterize the time consistency by a cocycle condition for the minimal penalty function. Taking advantage of this cocycle condition, we introduce a new methodology for the construction of time-consistent dynamic risk measures. Starting with BMO martingales, we provide new classes of time-consistent dynamic risk measures. These families generalize those obtained from backward stochastic differential equations. Quite importantly, starting with right-continuous BMO martingales, this construction naturally leads to paths with jumps.      

Set-valued risk measures as backward stochastic difference inclusions and equations

Scalar dynamic risk measures for univariate positions in continuous time are commonly represented via backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of set-valued functionals in the recent literature. There are two possible extensions of scalar backward stochastic differential equations for the set-valued framework: (1) backward stochastic differential inclusions, which evaluate the risk dynamics on the selectors of acceptable capital allocations; or (2) set-valued backward stochastic differential equations, which evaluate the risk dynamics on the full set of acceptable capital allocations as a singular object. In this work, the discrete-time setting is investigated with difference inclusions and difference equations in order to provide insights for such differential representations for set-valued dynamic risk measures in continuous time.

     

Vector-valued coherent risk measures

We define (d,n)-coherent risk measures as set-valued maps from into satisfying 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 from valued random portfolio to valued measure of risk. Necessary and sufficient conditions of coherent aggregation are provided.Received: February 2004, Mathematics Subject Classification (2000): 91B30, 46E30JEL Classification: D81, G31     

Robustness and sensitivity analysis of risk measurement procedures

Measuring the risk of a financial portfolio involves two steps: estimating the loss distribution of the portfolio from available observations and computing a ‘risk measure’ that summarizes the risk of the portfolio. We define the notion of ‘risk measurement procedure’, which includes both of these steps, and introduce a rigorous framework for studying the robustness of risk measurement procedures and their sensitivity to changes in the data set. Our results point to a conflict between the subadditivity and robustness of risk measurement procedures and show that the same risk measure may exhibit quite different sensitivities depending on the estimation procedure used. Our results illustrate, in particular, that using recently proposed risk measures such as CVaR/expected shortfall leads to a less robust risk measurement procedure than historical Value-at-Risk. We also propose alternative risk measurement procedures that possess the robustness property.     

On risk management problems related to a coherence property

Value at Risk has lost the battle against Expected Shortfall on theoretical grounds, the latter satisfying all coherence properties while the former may, on carefully constructed cases, lack the sub-additivity property that is in a sense, the most important property a risk measure ought to satisfy. While the superiority of Expected Shortfall is evident as a theoretical tool, little has been researched on the properties of estimators proposed in the literature. Since those estimators are the real tools for calculating bank capital reserves in practice, the natural question that one may ask is whether a given estimator of Expected Shortfall also satisfies the coherence properties. In this paper, we show that it is possible to have estimators of Expected Shortfall that do not satisfy the sub-additivity condition. This finding should motivate risk managers and quantitative asset managers to investigate further the properties of the estimators of the risk measures they are currently utilizing.     

An axiomatic characterization of capital allocations of coherent risk measures

An axiomatic definition of coherent capital allocations is given. It is shown that coherent capital allocations defined by the proposed axiom system are closely linked to coherent risk measures. More precisely, the associated risk measure of a coherent capital allocation is coherent and, conversely, for every coherent risk measure there exists a coherent capital allocation.     

A note on generalized distortion risk measures

A generalized distortion risk measure is introduced as power of the mean absolute deviation power of a distorted random variable with respect to a location parameter. This class of risk measures extends both the distortion risk measure by Wang and Denneberg and the class of financial risk measures by Pedersen and Satchell, which itself contains the class of Stone. Integral representations and a stop–loss order preserving property of a special up-side risk measure are derived.     

Methods of Partial Hedging

In this article we survey methods of dealing with the following problem: A financial agent is trying to hedge a claim C, without having enough initial capital to perform a perfect (super) replication. In particular, we describe results for minimizing the expected loss of hedging the claim C both in complete and incomplete continuous-time financial market models, and for maximizing the probability of perfect hedge in complete markets and markets with partial information. In these cases, the optimal strategy is in the form of a binary option on C, depending on the Radon-Nikodym derivative of the equivalent martingale measure which is optimal for a corresponding dual problem. We also present results on dynamic measures for the risk associated with the liability C, defined as the supremum over different scenarios of the minimal expected loss of hedging C. This revised version was published online in August 2006 with corrections to the Cover Date.     

Forward-looking portfolio selection with multivariate non-Gaussian models

In this study, we suggest a portfolio selection framework based on time series of stock log-returns, option-implied information, and multivariate non-Gaussian processes. We empirically assess a multivariate extension of the normal tempered stable (NTS) model and of the generalized hyperbolic (GH) one by implementing an estimation method that simultaneously calibrates the multivariate time series of log-returns and, for each margin, the univariate observed one-month implied volatility smile. To extract option-implied information, the connection between the historical measure P and the risk-neutral measure Q, needed to price options, is provided by the multivariate Esscher transform. The method is applied to fit a 50-dimensional series of stock returns, to evaluate widely known portfolio risk measures and to perform a forward-looking portfolio selection analysis. The proposed models are able to produce asymmetries, heavy tails, both linear and non-linear dependence and, to calibrate them, there is no need for liquid multivariate derivative quotes.     

Conditional risk measures in a bipartite market structure

In this paper, we study the effect of network structure between agents and objects on measures for systemic risk. We model the influence of sharing large exogeneous losses to the financial or (re)insurance market by a bipartite graph. Using Pareto-tailed losses and multivariate regular variation, we obtain asymptotic results for conditional risk measures based on the Value-at-Risk and the Conditional Tail Expectation. These results allow us to assess the influence of an individual institution on the systemic or market risk and vice versa through a collection of conditional risk measures. For large markets, Poisson approximations of the relevant constants are provided. Differences of the conditional risk measures for an underlying homogeneous and inhomogeneous random graph are illustrated by simulations.     

Decision making with Expected Shortfall and spectral risk measures: The problem of comparative risk aversion

We analyze spectral risk measures with respect to comparative risk aversion following Arrow (1965) and Pratt (1964) for deterministic wealth, and Ross (1981) for stochastic wealth. We argue that the Arrow–Pratt-concept per se well matches with economic intuition in standard financial decision problems, such as willingness to pay for insurance and simple portfolio problems. Different from the literature, we find that the widely-applied spectral Arrow–Pratt-measure is not a consistent measure of Arrow–Pratt-risk aversion. Instead, the difference between the antiderivatives of the corresponding risk spectra is valid. Within the framework of Ross, we show that the popular subclasses of Expected Shortfall, and exponential and power spectral risk measures cannot be completely ordered with respect to Ross-risk aversion. Thus, for all these subclasses, the concept of Ross-risk aversion is not generally compatible with Arrow–Pratt-risk aversion, but induces counter-intuitive comparative statics of its own. Compatibility can be achieved if asset returns are jointly normally distributed. The general lesson is that these restrictions have to be considered before spectral risk measures can be applied for the purpose of optimal decision making and regulatory issues.     

Spectral Risk Measures: Properties and Limitations

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.

Coherent and convex monetary risk measures for unbounded càdlàg processes

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.Due to errors during the typesetting process, this article was published incorrectly in Finance Stoch 9(3):369–387 (2005). The address of the first author was printed incorrectly, and in the whole paper the angular brackets were misprinted as [ ]. The complete corrected article is given here. The online version of the original paper can be found at: http://dx.doi.org/10.1007/s00780-004-0150-7     

Optional decomposition and Lagrange multipliers

Let be the set of equivalent martingale measures for a given process , and let be a process which is a local supermartingale with respect to any measure in . The optional decomposition theorem for states that there exists a predictable integrand such that the difference is a decreasing process. In this paper we give a new proof which uses techniques from stochastic calculus rather than functional analysis, and which removes any boundedness assumption.     

Two-sided coherent risk measures and their application in realistic portfolio optimization

By using a different derivation scheme, a new class of two-sided coherent risk measures is constructed in this paper. Different from existing coherent risk measures, both positive and negative deviations from the expected return are considered in the new measure simultaneously but differently. This innovation makes it easy to reasonably describe and control the asymmetry and fat-tail characteristics of the loss distribution and to properly reflect the investor’s risk attitude. With its easy computation of the new risk measure, a realistic portfolio selection model is established by taking into account typical market frictions such as taxes, transaction costs, and value constraints. Empirical results demonstrate that our new portfolio selection model can not only suitably reflect the impact of different trading constraints, but find more robust optimal portfolios, which are better than the optimal portfolio obtained under the conditional value-at-risk measure in terms of diversification and typical performance ratios.     

Remarks on a copula-based conditional value at risk for the portfolio problem

We deal with a multivariate conditional value at risk. Compared with the usual notion for the single random variable, a multivariate value at risk is concerned with several variables, and thus, the relation between each risk factor should be considered. We here introduce a new definition of copula-based conditional value at risk, which is real valued and ready to be computed. Copulas are known to provide a flexible method for handling a possible nonlinear structure; therefore, copulas may be naturally involved in the theory of value at risk. We derive a formula of our copula-based conditional value at risk in the case of Archimedean copulas, whose effectiveness is shown by examples. Numerical studies are also carried out with real data, which can be verified with analytical results.