The limitations of comonotonic additive risk measures: a literature review

The theory of risk measures has grown enormously in the last twenty years. In particular, risk measures satisfying the axiom of comonotonic additivity were extensively studied, arguably because of the affluence of results indicating interesting aspects of such risk measures. Recent research, however, has shown that this axiom is incompatible with properties that are central in specific contexts. In this paper we present a literature review of these incompatibilities. As a secondary contribution, we show that the comonotonic additivity axiom conflicts with the property of excess invariance for risk measures and, in a milder form, with the property of surplus invariance for acceptance sets

Three studies on risk measures : a focus on the comonotonic additivity property

The theory of risk measures has grown enormously in the last twenty years. In particular, risk measures satisfying the axiom of comonotonic additivity were extensively studied, arguably because of the affluence of results indicating interesting aspects of such risk measures. Recent research, however, has shown that this axiom is incompatible with properties that are central in specific contexts. In this paper we present a literature review of these incompatibilities. As a secondary contribution, we show that the comonotonic additivity axiom conflicts with the property of excess invariance for risk measures and, in a milder form, with the property of surplus invariance for acceptance sets. An elementary fact in the theory of risk measures is that acceptance sets induce risk measures and vice-versa. We present simple and yet general conditions on the acceptance sets under which their induced risk measures are comonotonic additive. With this result, we believe to fill a gap in the literature linking the properties of acceptance sets and risk measures: we show that acceptance sets induce comonotonic additive risk measures if the acceptance sets and their complements are stable under convex combinations of comonotonic random variables. As an extension of our results, we obtain a set of axioms on acceptance sets that allows one to induce risk measures that are additive for a priori chosen classes of random variables. Examples of such classes that were previously considered in the literature are independent random variables, uncorrelated random variables, and notably, comonotonic random variables. Taking investment decisions requires managers to consider how the current portfolio would be affected by the inclusion of other assets. In particular, it is of interest to know if adding a given asset would increase or decrease the risk of the current portfolio. However, this addition may reduce or increase the risk, depending on the risk measure being used. Arguably, risk sub-estimation is a major concern to regulatory agencies, and possibly to the financial firms themselves. To provide a more decisive and conservative conclusion about the effect of an additional asset on the risk of the current portfolio, we propose to assess this effect through the family of monetary risk measures that are consistent with second-degree stochastic dominance (SSD-consistent risk measures). This criterion provides a tool to identify financial positions that reduce the risk of the current portfolio, according to all monetary SSD-consistent risk measures. Also, this tool measures the smallest amount of money (the cost) necessary to turn the financial positions into risk reducers for the original portfolio. We characterize the cost of robust risk reduction through a monetary risk measure, a monetary acceptance set, the family of average values at risk, and through the infimum of the certainty equivalents of risk-averse agents with random initial wealth

Fundamental theorem of asset pricing with acceptable risk in markets with frictions

We study the range of prices at which a rational agent should contemplate transacting a financial contract outside a given securities market. Trading is subject to nonproportional transaction costs and portfolio constraints and full replication by way of market instruments is not always possible. Rationality is defined in terms of consistency with market prices and acceptable risk thresholds. We obtain a direct and a dual description of market-consistent prices with acceptable risk. The dual characterization requires an appropriate extension of the classical Fundamental Theorem of Asset Pricing where the role of arbitrage opportunities is played by acceptable deals, i.e., costless investment opportunities with acceptable risk-reward tradeoff. In particular, we highlight the importance of scalable acceptable deals, i.e., investment opportunities that are acceptable deals regardless of their volume.Comment: 28 page

One axiom to rule them all: A minimalist axiomatization of quantiles

We offer a minimalist axiomatization of quantiles among all real-valued mappings on a general set of distributions through only one axiom. This axiom is called ordinality: Quantiles are the only mappings that commute with all increasing and continuous transforms. Other convenient properties of quantiles—monotonicity, semicontinuity, comonotonic additivity, elicitability, and locality in particular—follow from this axiom. Furthermore, on the set of convexly supported distributions, the median is the only mapping that commutates with all monotone and continuous transforms. On a general set of distributions, the median interval is pinned down as the unique minimal interval-valued mapping that commutes with all monotone and continuous transforms. Finally, our main result, put in a decision-theoretic setting, leads to a minimalist axiomatization of quantile preferences. In banking and insurance, quantiles are known as the standard regulatory risk measure Value-at-Risk (VaR), and thus an axiomatization of VaR is obtained with only one axiom among law-based risk measures

One axiom to rule them all: An axiomatization of quantiles

We offer an axiomatic characterization of quantiles through only one axiom. Among all real-valued mappings on a general set of distributions, left quantiles are the only ones satisfying left-ordinal covariance, meaning that they commute with increasing left-continuous transforms; the case of right quantiles is analogous. Other convenient properties of quantiles, monotonicity in particular, follow from this axiom. In banking and insurance, quantiles are known as Value-at-Risk (VaR), a standard regulatory risk measure. Thus, we obtain an axiomatization of VaR with only one axiom among law-based risk measures. We further show that VaR can be alternatively characterized via the axiom of locality, plus four standard axioms relevant in financial risk management, namely, monotonicity, normalization, cash additivity, and semicontinuit

Innovations in Quantitative Risk Management

Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science

Innovations in Quantitative Risk Management

Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science

From Mirror to Mirage: The Idea of Logical Space in Kant, Wittgenstein, and van Fraassen

This dissertation investigates the origin, intellectual development and use of a semantic variant of the idea of logical space found implicitly in Kant and explicitly in early Wittgenstein and van Fraassen. It elucidates the idea of logical space as the idea of images or pictures representative of reality organized into a logico-mathematical structure circumscribing a form of all possible worlds. Its main claim is that application of these images or pictures to reality is through a certain conception of self. The first chapter presents a novel interpretation of Kant’s semantic theory of schemata in the Critique of Pure Reason, showing that a structure of the imagination induced by the transcendental self informs an implicit idea of logical space. The second chapter offers an intellectual history of the idea through developments in the organization of images introduced by Helmholtz and Hertz. The third chapter reveals early Wittgenstein’s idea of logical space to be his notion of the self, demonstrating how this serves to unify propositions of the Tractatus Logico-Philosophicus concerning solipsism, realism, ethics, aesthetics and mysticism with those pertaining to the picture theory of meaning. The fourth chapter provides a historical overview of the development of van Fraassen’s empiricism in relation to his adaptation of logical space, and evaluates his recent proposal in Scientific Representation: Paradoxes of Perspective that the problem of coordination in the semantic view of theories is dissolved through self-location in logical space. After identifying a number of problems this proposal creates for his empiricism, a brief suggestion is made about how van Fraassen might improve upon his conception of logical space, and how an empiricist view of scientific representation might be understood as a result

Geobase Information System Impacts on Space Image Formats

As Geobase Information Systems increase in number, size and complexity, the format compatability of satellite remote sensing data becomes increasingly more important. Because of the vast and continually increasing quantity of data available from remote sensing systems the utility of these data is increasingly dependent on the degree to which their formats facilitate, or hinder, their incorporation into Geobase Information Systems. To merge satellite data into a geobase system requires that they both have a compatible geographic referencing system. Greater acceptance of satellite data by the user community will be facilitated if the data are in a form which most readily corresponds to existing geobase data structures. The conference addressed a number of specific topics and made recommendations