# The strictest common relaxation of a family of risk measures

Berend Roorda, J.M. Schumacher

1 Citation (Scopus)

### Abstract

Operations which form new risk measures from a collection of given (often simpler) risk measures have been used extensively in the literature. Examples include convex combination, convolution, and the worst-case operator. Here we study the risk measure that is constructed from a family of given risk measures by the best-case operator; that is, the newly constructed risk measure is defined as the one that is as restrictive as possible under the condition that it accepts all positions that are accepted under any of the risk measures from the family. In fact we define this operation for conditional risk measures, to allow a multiperiod setting. We show that the well-known VaR risk measure can be constructed from a family of conditional expectations by a combination that involves both worst-case and best-case operations. We provide an explicit description of the acceptance set of the conditional risk measure that is obtained as the strictest common relaxation of two given conditional risk measures.
Original language English 29-34 Insurance: mathematics & economics 48 1 https://doi.org/10.1016/j.insmatheco.2010.09.001 Published - 2011

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Risk Measures
Family
Measure of risk
Risk measures
Convex Combination
Conditional Expectation
Operator
Convolution

• METIS-268570
• IR-104082

### Cite this

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abstract = "Operations which form new risk measures from a collection of given (often simpler) risk measures have been used extensively in the literature. Examples include convex combination, convolution, and the worst-case operator. Here we study the risk measure that is constructed from a family of given risk measures by the best-case operator; that is, the newly constructed risk measure is defined as the one that is as restrictive as possible under the condition that it accepts all positions that are accepted under any of the risk measures from the family. In fact we define this operation for conditional risk measures, to allow a multiperiod setting. We show that the well-known VaR risk measure can be constructed from a family of conditional expectations by a combination that involves both worst-case and best-case operations. We provide an explicit description of the acceptance set of the conditional risk measure that is obtained as the strictest common relaxation of two given conditional risk measures.",
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The strictest common relaxation of a family of risk measures. / Roorda, Berend; Schumacher, J.M.

In: Insurance: mathematics & economics, Vol. 48, No. 1, 2011, p. 29-34.

TY - JOUR

T1 - The strictest common relaxation of a family of risk measures

AU - Roorda, Berend

AU - Schumacher, J.M.

PY - 2011

Y1 - 2011

N2 - Operations which form new risk measures from a collection of given (often simpler) risk measures have been used extensively in the literature. Examples include convex combination, convolution, and the worst-case operator. Here we study the risk measure that is constructed from a family of given risk measures by the best-case operator; that is, the newly constructed risk measure is defined as the one that is as restrictive as possible under the condition that it accepts all positions that are accepted under any of the risk measures from the family. In fact we define this operation for conditional risk measures, to allow a multiperiod setting. We show that the well-known VaR risk measure can be constructed from a family of conditional expectations by a combination that involves both worst-case and best-case operations. We provide an explicit description of the acceptance set of the conditional risk measure that is obtained as the strictest common relaxation of two given conditional risk measures.

AB - Operations which form new risk measures from a collection of given (often simpler) risk measures have been used extensively in the literature. Examples include convex combination, convolution, and the worst-case operator. Here we study the risk measure that is constructed from a family of given risk measures by the best-case operator; that is, the newly constructed risk measure is defined as the one that is as restrictive as possible under the condition that it accepts all positions that are accepted under any of the risk measures from the family. In fact we define this operation for conditional risk measures, to allow a multiperiod setting. We show that the well-known VaR risk measure can be constructed from a family of conditional expectations by a combination that involves both worst-case and best-case operations. We provide an explicit description of the acceptance set of the conditional risk measure that is obtained as the strictest common relaxation of two given conditional risk measures.

KW - METIS-268570

KW - IR-104082

U2 - 10.1016/j.insmatheco.2010.09.001

DO - 10.1016/j.insmatheco.2010.09.001

M3 - Article

VL - 48

SP - 29

EP - 34

JO - Insurance: mathematics & economics

JF - Insurance: mathematics & economics

SN - 0167-6687

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