Update rules for convex risk measures

In the first part of the paper we investigate the properties that describe the intertemporal structure of dynamic convex risk measures. The usual backward approach to dynamic risk assessment leads to strong and weak versions of time consistency. As an alternative, we introduce a forward approach of consecutivity. In the second part we discuss the problem of how to update a convex risk measure when new information arrives. We analyse to what extent the above properties are appropriate update criteria.     

General convex order on risk aggregation

Using a general notion of convex order, we derive general lower bounds for risk measures of aggregated positions under dependence uncertainty, and this in arbitrary dimensions and for heterogeneous models. We also prove sharpness of the bounds obtained when each marginal distribution has a decreasing density. The main result answers a long-standing open question and yields an insight in optimal dependence structures. A numerical algorithm provides bounds for quantities of interest in risk management. Furthermore, our numerical results suggest that the bounds obtained in this paper are generally sharp for a broader class of models.     

Satisfying convex risk limits by trading

A random variable, representing the final position of a trading strategy, is deemed acceptable if under each of a variety of probability measures its expectation dominates a floor associated with the measure. The set of random variables representing pre-final positions from which it is possible to trade to final acceptability is characterized. In particular, the set of initial capitals from which one can trade to final acceptability is shown to be a closed half-line . Methods for computing are provided, and the application of these ideas to derivative security pricing is developed.Received: May 2004, Mathematics Subject Classification (2000): 91B30, 60H30, 60G44JEL Classification: G10Steven E. Shreve: Work supported by the National Science Foundation under grants DMS-0103814 and DMS-0139911.Reha Tütüncü: Work supported by National Science Foundation under grants CCR-9875559 and DMS-0139911.     

Optimal reinsurance under general law-invariant risk measures

In recent years, general risk measures play an important role in risk management in both finance and insurance industry. As a consequence, there is an increasing number of research on optimal reinsurance decision problems using risk measures beyond the classical expected utility framework. In this paper, we first show that the stop-loss reinsurance is an optimal contract under law-invariant convex risk measures via a new simple geometric argument. A similar approach is then used to tackle the same optimal reinsurance problem under Value at Risk and Conditional Tail Expectation; it is interesting to note that, instead of stop-loss reinsurances, insurance layers serve as the optimal solution. These two results highlight that law-invariant convex risk measure is better and more robust, in the sense that the corresponding optimal reinsurance still provides the protection coverage against extreme loss irrespective to the potential increment of its probability of occurrence, to expected larger claim than Value at Risk and Conditional Tail Expectation which are more commonly used. Several illustrative examples will be provided.     

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     

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.      

Robust portfolio selection under downside risk measures

We investigate a robust version of the portfolio selection problem under a risk measure based on the lower-partial moment (LPM), where uncertainty exists in the underlying distribution. We demonstrate that the problem formulations for robust portfolio selection based on the worst-case LPMs of degree 0, 1 and 2 under various structures of uncertainty can be cast as mathematically tractable optimization problems, such as linear programs, second-order cone programs or semidefinite programs. We perform extensive numerical studies using real market data to reveal important properties of several aspects of robust portfolio selection. We can conclude from our results that robustness does not necessarily imply a conservative policy and is indeed indispensable and valuable in portfolio selection.     

Convex risk measures for the aggregation of multiple information sources and applications in insurance

We propose a novel class of convex risk measures, based on the concept of the Fréchet mean, designed in order to handle uncertainty which arises from multiple information sources regarding the risk factors of interest. The proposed risk measures robustly characterize the exposure of the firm, by filtering out appropriately the partial information available in individual sources into an aggregate model for the risk factors of interest. Importantly, the proposed risks can be expressed in closed analytic forms allowing for interesting qualitative interpretations as well as comparative statics and thus facilitate their use in the everyday risk management process of the insurance firms. The potential use of the proposed risk measures in insurance is illustrated by two concrete applications, capital risk allocation and premia calculation under uncertainty.     

A supermartingale relation for multivariate risk measures

The equivalence between multiportfolio time consistency of a dynamic multivariate risk measure and a supermartingale property is proven. Furthermore, the dual variables under which this set-valued supermartingale is a martingale are characterized as the worst-case dual variables in the dual representation of the risk measure. Examples of multivariate risk measures satisfying the supermartingale property are given. Crucial for obtaining the results are dual representations of scalarizations of set-valued dynamic risk measures, which are of independent interest in the fast growing literature on multivariate risks.     

Optimal capital and risk allocations for law- and cash-invariant convex functions

In this paper we provide a complete solution to the existence and characterization problem of optimal capital and risk allocations for not necessarily monotone, law-invariant convex risk measures on the model space L ^p for any p∈[1,∞]. Our main result says that the capital and risk allocation problem always admits a solution via contracts whose payoffs are defined as increasing Lipschitz-continuous functions of the aggregate risk. Filipović is supported by WWTF (Vienna Science and Technology Fund). Svindland gratefully acknowledges financial support from Munich Re Grant for doctoral students and hospitality of the Research Unit of Financial and Actuarial Mathematics, Vienna University of Technology. We thank Beatrice Acciaio and Walter Schachermayer for fruitful discussions and an anonymous referee for helpful remarks.     

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.     

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.     

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.