Abstract
From a theoretical point of view if we are interested in measuring the risk faced by financial institutions, a specific value has to be provided. In this chapter, extensions of both risk measures and approaches presented in Chapters 3 and 4 are presented.
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Notes
- 1.
Law-invariance: for any combination of risks X and Y with respective cumulative distribution functions F(x) and F(y), if F(x) = F(y), then ρ(X) = ρ(Y ); comonotonic additivity: for every comonotonic random variables X and Y , ρ(X + Y ) = ρ(X) + ρ(Y ).
- 2.
X (m) is also called the mth order statistic, which is a fundamental tool in nonparametric statistics.
- 3.
These distributions represent financial data with different features.
- 4.
The results of A-D statistic are similar to those obtained with the K-S statistic, so we do not provide them in the paper.
- 5.
Both integrals in (5.1.13) are well defined and take a value in [0, +∞]. Provided that at least one of the two integrals is finite, the distorted expectation ρ g(X) is well defined and takes a value in [−∞, +∞].
- 6.
The distortion risk measure is a special class of the so-called Choquet expected utility (Bassett Jr et al. 2004), i.e., the expected utility calculated under a modified probability measure.
- 7.
For example, if we consider the so-called likelihood depth, the depth function is simply the probability density. Otherwise, the Mahalanobis depth (Mahalanobis 1936), D M(y, G), of a vector y with respect to a multivariate probability distribution G is defined by
$$\displaystyle \begin{aligned}D^M(y,G)=\left[1+(y-\mu_G)'\Sigma_G^{-1}(y-\mu_G)\right]^{-1},\end{aligned}$$where μ G and ΣG are respectively the mean vector and the covariance matrix of a random vector of distribution G.
- 8.
The centre is also sometimes called median of the distribution, but it is quite different from a statistical median in the sense of a 0.5-quantile.
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Guégan, D., Hassani, B.K. (2019). Extensions for Risk Measures: Univariate and Multivariate Approaches. In: Risk Measurement. Springer, Cham. https://doi.org/10.1007/978-3-030-02680-6_5
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