The intuition of risk is based on two main concepts: loss and variability. In this paper, we present a composition of risk and deviation measures, which contemplate these two concepts. Based on the proposed Limitedness axiom, we prove that this resulting composition, based on properties of the two components, is a coherent risk measure. Similar results for the cases of convex and co-monotone risk measures are exposed. We also provide examples of known and new risk measures constructed under this framework in order to highlight the importance of our approach, especially the role of the Limitedness axiom.
This article aims to verify whether the investments in companies with better practices of Corporate Social Responsibility, Corporate Governance and Sustainability presents performance differences in relation to investments in companies that represent the market. To this end, we analyzed the series of daily returns of Ibovespa, IGC, IGCT, ISE and ITAG through GARCH model and mean differences tests. The results show that the conditional volatility of the differentiated practices indexes is significantly smaller than the Ibovespa volatility, although the correlation between the returns is very high. Besides that, the Sharpe Index (1966) of the better practices indexes shows that their return per unit of risk is significantly higher than Ibovespa's. Thus, in general, it is possible to conclude that investments in different practices constitute a less risky and more profitable alternative to the investor.
The aim of this paper is to introduce a risk measure that extends the Gini-type measures of risk and variability, the Extended Gini Shortfall, by taking risk aversion into consideration. Our risk measure is coherent and catches variability, an important concept for risk management. The analysis is made under the Choquet integral representations framework. We expose results for analytic computation under well-known distribution functions. Furthermore, we provide a practical application.
JEL classification: C6, G10a prominent trend associated with tail-based risk measures has emerged, especially with the most popular ones nowadays: the Value-at-Risk (VaR) and the Expected Shortfall (ES). However, this kind of risk measures does not capture the variability of a financial position, a primitive but relevant concept. In order to solution this issue, some authors have proposed and studied specific examples of risk measures.In this sense, Fischer  considered combining the mean and semi-deviations. Regarding tail risk, Furman and Landsman  proposed a measure that weighs the mean and standard deviation in the truncated tail by VaR, while Righi and Ceretta  considered penalizing the ES by the dispersion of losses exceeding it. From a practical perspective, Righi and Borenstein  explored this concept, calling the approach as loss-deviation, for portfolio optimization. In a more general fashion, Righi  presents results and examples about compositions of risk and variability measures in order to ensure solid theoretical properties.Recently, Furman et al.  introduced the Gini Shortfall (GS) risk measure which is coherent and satisfies co-monotonic additivity. GS is a composition between ES and tail based Gini coefficient. However, GS supposes that all individuals have the same attitude towards risk, while agents differ in the way they take personal decisions that involve risk because of discrepancies in their risk aversion. To incorporate such psychological behavior in tail risk analysis, we introduce a generalized version of the GS. This risk measure, called Extended Gini Shortfall (EGS), captures the notion of variability, satisfies the co-monotonic additivity property, and it is coherent under a necessary and sufficient condition for its loading parameter. The consideration of the decision-maker risk aversion, joined to these properties, is in consonance to what agents seek when searching for a suitable measure of risk. The approach followed in this article leads us to a new family of spectral risk measures, proposed by Acerbi , with an attractive weighting function.In this sense, we discuss, in a separated manner, the properties from the variability term and our composed risk measure. Moreover, we discuss in details the role of each parameter in the mentioned weighting function. Furthermore, we expose results on analytic formulations for computation of EGS under known distribution functions. Our focus in this paper is on theoretical results, but this approach gives rise to further forthcoming inve...
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