Risk measures based on target risk profiles

TL;DR

This paper introduces a flexible class of adjusted risk measures based on target risk profiles, generalizing the adjusted Expected Shortfall to better detect tail risks and satisfy desirable mathematical properties.

Contribution

It generalizes the concept of adjusted Expected Shortfall to a broader family of risk measures, providing theoretical analysis and dual representations, with practical testing on S&P 500 data.

Findings

Adjusted risk measures can satisfy positive homogeneity and subadditivity.

Theoretical conditions for desirable properties are established.

Performance demonstrated through a case study on S&P 500.

Abstract

We address the problem that classical risk measures may not detect the tail risk adequately. This can occur for instance due to averaging when calculating the Expected Shortfall. The current literature proposes the so-called adjusted Expected Shortfall as a solution. This risk measure is the supremum of Expected Shortfalls for all possible levels, adjusted with a function $g$, the so-called target risk profile. We generalize this idea by using a family of risk measures which allows for more choices than Expected Shortfalls, leading to the concept of adjusted risk measures. An adjusted risk measure quantifies the minimal amount of capital that has to added to a financial position $X$ to ensure that each risk measure out of the chosen family is smaller or equal to the target risk profile $g(p)$ for the corresponding level $p\in[0,1]$. We discuss a variety of assumptions such that desirable properties for risk measures are satisfied in this setup. From a theoretical point of view, our main contribution is the analysis of equivalent assumptions such that an adjusted risk measure is positive homogeneous and subadditive. Furthermore, we show that these conditions hold for several adjusted risk measures beyond the adjusted Expected Shortfall. In addition, we derive their dual representations. Finally, we test the performance in a case study based on the S$\&$P $500$.