# Seven Proofs for the Subadditivity of Expected Shortfall

Dependence Modeling (2015)

- Volume: 3, Issue: 1, page 126-140, electronic only
- ISSN: 2300-2298

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topPaul Embrechts, and Ruodu Wang. "Seven Proofs for the Subadditivity of Expected Shortfall." Dependence Modeling 3.1 (2015): 126-140, electronic only. <http://eudml.org/doc/275999>.

@article{PaulEmbrechts2015,

abstract = {Subadditivity is the key property which distinguishes the popular risk measures Value-at-Risk and Expected Shortfall (ES). In this paper we offer seven proofs of the subadditivity of ES, some found in the literature and some not. One of the main objectives of this paper is to provide a general guideline for instructors to teach the subadditivity of ES in a course. We discuss the merits and suggest appropriate contexts for each proof.With different proofs, different important properties of ES are revealed, such as its dual representation, optimization properties, continuity, consistency with convex order, and natural estimators.},

author = {Paul Embrechts, Ruodu Wang},

journal = {Dependence Modeling},

keywords = {Expected Shortfall; TVaR; subadditivity; comonotonicity; Value-at-Risk; risk management; education; expected shortfall; value-at-risk},

language = {eng},

number = {1},

pages = {126-140, electronic only},

title = {Seven Proofs for the Subadditivity of Expected Shortfall},

url = {http://eudml.org/doc/275999},

volume = {3},

year = {2015},

}

TY - JOUR

AU - Paul Embrechts

AU - Ruodu Wang

TI - Seven Proofs for the Subadditivity of Expected Shortfall

JO - Dependence Modeling

PY - 2015

VL - 3

IS - 1

SP - 126

EP - 140, electronic only

AB - Subadditivity is the key property which distinguishes the popular risk measures Value-at-Risk and Expected Shortfall (ES). In this paper we offer seven proofs of the subadditivity of ES, some found in the literature and some not. One of the main objectives of this paper is to provide a general guideline for instructors to teach the subadditivity of ES in a course. We discuss the merits and suggest appropriate contexts for each proof.With different proofs, different important properties of ES are revealed, such as its dual representation, optimization properties, continuity, consistency with convex order, and natural estimators.

LA - eng

KW - Expected Shortfall; TVaR; subadditivity; comonotonicity; Value-at-Risk; risk management; education; expected shortfall; value-at-risk

UR - http://eudml.org/doc/275999

ER -

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