Risk Measures and Comonotonicity: A Review

By Jan Dhaene, Steven Vanduffel, Marc J. Goovaerts, Rob Kaas, Qihe Tang, David Vyncke. 2006

337 citations recorded by the source corpus.

In this paper we examine and summarize properties of several well-known risk measures that can be used in the framework of setting solvency capital requirements for a risky business. Special attention is given to the class of (concave) distortion risk measures. We investigate the relationship between these risk measures and theories of choice under risk. Furthermore we consider the problem of how to evaluate risk measures for sums of non-independent random variables. Approximations for such sums, based on the concept of comonotonicity, are proposed. Several examples are provided to illustrate properties or to prove that certain properties do not hold. Although the paper contains several new results, it is written as an overview and pedagogical introduction to the subject of risk measurement. The paper is an extended version of Dhaene et al. [11] .

Subfields: Finance, Management Science and Operations Research, Evaluation & Benchmarks

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Citations in the corpus

Autoregressive Quantile Networks for Generative Modeling (2018, background): “Two random variables are comonotonic if and only if they can be expressed as non-decreasing (deterministic) functions of a single random variable (Dhaene et al., 2006).”
Optimal reinsurance designs based on risk measures: a review (2020, background): “See Dhaene et al. (2006) for a detailed discussion on the relationship between CVaR and CTE.”
Properties of Distortion Risk Measures (2009, background): “A general overview of distortion risk measures and their relation with orderings of risks and the concept of comonotonicity can be found in Dhaene et al. (2006) .”
Remarks on quantiles and distortion risk measures (2012, neutral): “Applying this formula to the distortion function defined in (5), we find that qg1⁄2X 1⁄4 F 1þ X ðpÞ: As FX(p) and FX(p) are in general not equal, we can indeed conclude that (6) is in general not valid for a l.”
Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles (2013, neutral): “Furthermore, G(θ−) = ∫ (1+θ)r(SX(t))−g(SX(t))≤0 r(SX(t))dt follows and (11) is proved.”
Risk aggregation with dependence uncertainty (2014, background): “Since it is well-known that the convex ordering upper bound of Sn(F1, · · · , Fn) is given by the comonotonic scenario of X, a sharp upper bound on ρ(S) over S ∈ Sn(F1, · · · , Fn) is ρ(nF−1(U)) where U ∼ U [0, 1] and it is well-discussed in the literature (for a review, see Dhaene et al. (2006)).”
Seven Proofs for the Subadditivity of Expected Shortfall (2015, methodology): “The idea of this proof is presented in [36] and is also used in the review paper [14].”
Model uncertainty and VaR aggregation (2013, background): “For a detailed discussion of the concept of comonotonicity within quantitative risk management we refer to Dhaene et al. (2002) and Dhaene et al. (2006); see also McNeil et al. (2005, Section 6.2.2).”
Optimality of general reinsurance contracts under CTE risk measure (2011, methodology): “For instance,many authors (see, for example, Landsman and Valdez, 2003 and Dhaene et al., 2006) have defined CTE as CTEα(Z) = E[Z |Z > VaRα(Z)].”
Tail Conditional Variance for Elliptically Contoured Distributions (2004, background): “Hence, it immediately follows that (5) coincides with (4).”
Value-at-Risk Bounds with Variance Constraints (2015, neutral): “Then, Y 6cx Z ⇔ u ∫ 0 F−1(p)dp > u ∫ 0 G−1(p)dp for all 0 < u < 1 (A.3) ⇔ F−1 ≺ S G−1, where ≺ S is the Schur order (see Rüschendorf (1983b; 2013, Corollary 3.26 page 430) and Dhaene et al. (2006)).”
Weighted Risk Capital Allocations (2008, neutral): “There is a great variety of pcp’s in the literature; see, e.g., Gerber (1979), Buhlmann (1980, 1984),Goovaertsetal.(1984),Denneberg(1994),Kaasetal.(1994), Wang (1995, 1996, 1998), Tsanakas and Desli (2003), Heilpern (2003), Young (2004), Denuit et al. (2005), Denuit et al. (2006), Dhaene et al. (20”
Optimal Capital Allocation Principles (2005, neutral): “See Acerbi and Tasche (2002) and Remark 4.2.3. in Dhaene et al. (2006).”
Convex order and comonotonic conditional mean risk sharing (2012, background): “Recall from Dhaene et al. (2006) that TVaR is a subadditive risk measure, which is additive for comonotonic risks.”
Optimal Reinsurance Revisited – A Geometric Approach (2010, background): “( (X X +( (X X +f a a a( ( )) ( ( )) ( ( ))fVaR VaR) )f f X- + -+) )I I = X VaR- 93216_Astin40_1_09.indd 228 11-05-2010 09:40:02 Moreover, we know from Theorem 2.1 of Dhaene et al. ( 2006 ) that ((X af +( (X f X)f f X a a a a a( ) ( )) ) ( ) ( ( )) ( ) . f p dp S p dp f a VaR VaR VaR) )0 0 1 - - = -”