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

Abstract

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.