Abstract
Spectral risk measures are defined as the most attractive subclass of coherent quantile-based risk measures, with a remarkable aptitude for concretizing the decision-maker's subjective attitude toward risk. This chapter raises the problem of underrepresentation of the subclass of spectral risk measures by reviewing the standard spectral risk measures proposed in the literature. In parallel, a discussion about the approaches behind the conception of these risk measures is held. Through this discussion, the authors spot a number of problems with each of these proposals that stand against the reliable applicability of these risk measures in practice.
TopIntroduction
In quantitative risk management, measuring risk is of crucial importance. For such purpose, a variety of risk measures have been proposed in the financial literature. Risk measures are quantitative tools that map financial positions, modelled by random variables, to capital amounts which serve as a hedging against the underlying potential risks. The authors refer the reader to (Pflug & Rӧmisch, 2007; Delbaen, 2012; McNeil et al., 2005; Fӧllmer & Schied, 2016) for a comprehensive review of the use of risk measures in modern risk management. In this sense, the development of risk measurement theory has been sustained with the proposition of axiomatic approaches for risk measures. Especially, after the pioneering work of (Artzner et al., 1999), who defined the class of coherent risk measures as risk measures that satisfy desirable properties that have been conceded by the modern risk theory, namely: Monotonicity, Translation Invariance, Sub-additivity and Positive Homogeneity.
Among coherent risk measures an attractive subclass, called spectral risk measures, has emerged. Spectral risk measures were primarily introduced, by (Acerbi, 2002), as an extension for Expected Shortfall (Acerbi & Tasche, 2002). Beside coherence, spectral risk measures are law-invariant and co-monotonically additive (Kusuoka, 2001). Yet, the substantial contribution brought by this subclass consists in considering the psychological behavior of the decision-maker in the risk measurement process. In fact, a spectral risk measure is characterized with a weighting function that attributes subjective weights to the potential outcomes of a financial position, according to the decision-maker’s attitude toward risk.
Due to its distinctive features, the subclass of spectral risk measures has been the subject of a wide scope of surveys and works, since its introduction. For instance, spectral risk measures have been studied in the framework of modern portfolio theory (e.g., (Acerbi & Tasche, 2002; Adam et al., 2008; Brandtner, 2013)), for futures cleaning house margin requirements (e.g., (Cotter & Dowd, 2006)), in the framework of Extreme Value Theory (e.g., (Fałdziński et al., 2012)) and on the problem of comparative risk-aversion (e.g., (Brandtner & Kürsten, 2015)), to name a few.
Key Terms in this Chapter
Variability Measure: A function that allows the quantification of the variability or dispersion of the realizations of a given random variable (modeling a financial position).
Admissible Spectrum: A weighting function that satisfies the three first conditions i)-iii) detailed above in the present chapter.
Weighting Function: For a given risky financial position, a weighting function associates to each realization a given weight which, in the context of the present chapter, reflects the investor’s psychological aspect regarding risk.
Utility Function: A mapping that evaluates the satisfaction brought by a specific realization for a given investor.
Expected Shortfall: The expected shortfall, at a given level of confidence a in the unit interval, is a measure of risk that represents the mean loss beyond the corresponding value-at-risk.
Risk Measure: A mapping that associates a real value to a given financial position, which represents the risk value.
Value-at-Risk: The value-at-risk, at a given level of confidence a in the unit interval, is a risk measure that coincides with the a-quantile of the underlying financial position.
Risk-Aversion: A quantitative variable that reflects the psychological behavior or attitude toward risk of a given investor.