Entropy Coherent and Entropy Convex Measures of Risk - Eurandom
Entropy Coherent and Entropy Convex Measures of Risk - Eurandom Entropy Coherent and Entropy Convex Measures of Risk - Eurandom
Question Rephrased [2] ◮ We consider in addition with with β : M → [0, 1]. ρ(X) = φ −1 (¯ρ(−φ(−X))), ¯ρ(X) = sup Q∈M⊂Q β(Q)EQ [−X] , ◮ Preferences of Chateauneuf and Faro (2010). ◮ With ¯ρ as given above, negative certainty equivalents are invariant under positive multiplication of u (or φ); complementary case. ◮ β : M → [0,1] can be viewed as a discount factor; ¯ρ seems natural. ◮ Includes multiple priors preferences as a special case. ◮ Recall question: find sufficient and necessary conditions under which ρ (not ¯ρ) is a convex risk measure. Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 12/40
Results [1] The contribution of this paper is twofold. ◮ First we derive precise connections between risk measurement under the theories of variational, homothetic and multiple priors preferences on the one hand and risk measurement using convex measures of risk on the other. ◮ This is, despite the vast literature on both paradigms, a hitherto open problem. ◮ In particular, we identify two subclasses of convex risk measures that we call entropy coherent and entropy convex measures of risk, and that include all coherent risk measures. ◮ We show that, under technical conditions, negative certainty equivalents under variational, homothetic, and multiple priors preferences are translation invariant if and only if they are convex, entropy convex, and entropy coherent measures of risk, respectively. Entropy Coherent and Entropy Convex Measures of Risk Advances in Financial Mathematics, Eurandom, Eindhoven 13/40
- Page 1 and 2: Entropy Coherent and Entropy Convex
- Page 3 and 4: Convex Measures of Risk ◮ Convex
- Page 5 and 6: Multiple Priors Preferences ◮ A s
- Page 7 and 8: Homothetic Preferences ◮ Recently
- Page 9 and 10: Variational and Homothetic Preferen
- Page 11: Question Rephrased [1] ◮ In other
- Page 15 and 16: Results [3] ◮ These connections s
- Page 17 and 18: 2. Entropic Measures of Risk [1] We
- Page 19 and 20: Two Interpretations 1. Kullback-Lei
- Page 21 and 22: Entropy Coherence and Entropy Conve
- Page 23 and 24: Again Two Interpretations [2] The e
- Page 25 and 26: A Basic Duality Result Define and T
- Page 27 and 28: 3. Characterization Results [1] Rec
- Page 29 and 30: Remark 1 ◮ The case that ρ is en
- Page 31 and 32: Characterization Results [3] Recall
- Page 33 and 34: 4. Duality Results [1] Recall that
- Page 35 and 36: 5. Distribution Invariant Represent
- Page 37 and 38: 6.1 Risk Sharing ◮ Suppose that t
- Page 39 and 40: Portfolio Optimization and Indiffer
Results [1]<br />
The contribution <strong>of</strong> this paper is tw<strong>of</strong>old.<br />
◮ First we derive precise connections between risk measurement under the<br />
theories <strong>of</strong> variational, homothetic <strong>and</strong> multiple priors preferences on the<br />
one h<strong>and</strong> <strong>and</strong> risk measurement using convex measures <strong>of</strong> risk on the<br />
other.<br />
◮ This is, despite the vast literature on both paradigms, a hitherto open<br />
problem.<br />
◮ In particular, we identify two subclasses <strong>of</strong> convex risk measures that we<br />
call entropy coherent <strong>and</strong> entropy convex measures <strong>of</strong> risk, <strong>and</strong> that<br />
include all coherent risk measures.<br />
◮ We show that, under technical conditions, negative certainty equivalents<br />
under variational, homothetic, <strong>and</strong> multiple priors preferences are<br />
translation invariant if <strong>and</strong> only if they are convex, entropy convex, <strong>and</strong><br />
entropy coherent measures <strong>of</strong> risk, respectively.<br />
<strong>Entropy</strong> <strong>Coherent</strong> <strong>and</strong> <strong>Entropy</strong> <strong>Convex</strong> <strong>Measures</strong> <strong>of</strong> <strong>Risk</strong> Advances in Financial Mathematics, Eur<strong>and</strong>om, Eindhoven 13/40
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