A Generalization of the Aumann-Shapley Value for Risk Capital Allocation Problems

Research output: Working paperDiscussion paperOther research output

394 Downloads (Pure)

Abstract

Abstract: This paper analyzes risk capital allocation problems. For risk capital allocation problems, the aim is to allocate the risk capital of a firm to its divisions. Risk capital allocation is of central importance in risk-based performance measurement. We consider a case in which the aggregate risk capital is determined via a coherent risk measure. The academic literature advocates an allocation rule that, in game-theoretic terms, is equivalent to using the Aumann-Shapley value as solution concept. This value is however not well-defined in case a differentiability condition is not satisfied. As an alternative, we introduce an allocation rule inspired by the Shapley value in a fuzzy setting. We take a grid on a fuzzy participation set, define paths on this grid and construct an allocation rule based on a path. Then, we define a rule as the limit of the average over these allocations, when the grid size converges to zero. We introduce this rule for a broad class of coherent risk measures. We show that if the Aumann-Shapley value is well-defined, the allocation rule coincides with it. If the Aumann-Shapley value is not defined, which is due to non-differentiability problems, the allocation rule specifies an explicit allocation. It corresponds with the Mertens value, which is originally characterized in an axiomatic way (Mertens, 1988), whereas we provide an asymptotic argument.
Original languageEnglish
Place of PublicationTilburg
PublisherEconometrics
Number of pages43
Volume2012-091
Publication statusPublished - 2012

Publication series

NameCentER Discussion Paper
Volume2012-091

Fingerprint

Capital allocation
Risk capital
Shapley value
Allocation problem
Allocation rules
Grid
Participation
Axiomatics
Coherent measures of risk
Differentiability
Performance measurement
Rule-based
Coherent risk measures
Solution concepts

Keywords

  • capital allocation
  • risk capital
  • Aumann-Shapley value
  • non-differentiability
  • fuzzy games

Cite this

Boonen, T. J., De Waegenaere, A. M. B., & Norde, H. W. (2012). A Generalization of the Aumann-Shapley Value for Risk Capital Allocation Problems. (CentER Discussion Paper; Vol. 2012-091). Tilburg: Econometrics.
@techreport{2c502ef876f047f5ab451833b5f41103,
title = "A Generalization of the Aumann-Shapley Value for Risk Capital Allocation Problems",
abstract = "Abstract: This paper analyzes risk capital allocation problems. For risk capital allocation problems, the aim is to allocate the risk capital of a firm to its divisions. Risk capital allocation is of central importance in risk-based performance measurement. We consider a case in which the aggregate risk capital is determined via a coherent risk measure. The academic literature advocates an allocation rule that, in game-theoretic terms, is equivalent to using the Aumann-Shapley value as solution concept. This value is however not well-defined in case a differentiability condition is not satisfied. As an alternative, we introduce an allocation rule inspired by the Shapley value in a fuzzy setting. We take a grid on a fuzzy participation set, define paths on this grid and construct an allocation rule based on a path. Then, we define a rule as the limit of the average over these allocations, when the grid size converges to zero. We introduce this rule for a broad class of coherent risk measures. We show that if the Aumann-Shapley value is well-defined, the allocation rule coincides with it. If the Aumann-Shapley value is not defined, which is due to non-differentiability problems, the allocation rule specifies an explicit allocation. It corresponds with the Mertens value, which is originally characterized in an axiomatic way (Mertens, 1988), whereas we provide an asymptotic argument.",
keywords = "capital allocation, risk capital, Aumann-Shapley value, non-differentiability, fuzzy games",
author = "T.J. Boonen and {De Waegenaere}, A.M.B. and H.W. Norde",
note = "Pagination: 43",
year = "2012",
language = "English",
volume = "2012-091",
series = "CentER Discussion Paper",
publisher = "Econometrics",
type = "WorkingPaper",
institution = "Econometrics",

}

Boonen, TJ, De Waegenaere, AMB & Norde, HW 2012 'A Generalization of the Aumann-Shapley Value for Risk Capital Allocation Problems' CentER Discussion Paper, vol. 2012-091, Econometrics, Tilburg.

A Generalization of the Aumann-Shapley Value for Risk Capital Allocation Problems. / Boonen, T.J.; De Waegenaere, A.M.B.; Norde, H.W.

Tilburg : Econometrics, 2012. (CentER Discussion Paper; Vol. 2012-091).

Research output: Working paperDiscussion paperOther research output

TY - UNPB

T1 - A Generalization of the Aumann-Shapley Value for Risk Capital Allocation Problems

AU - Boonen, T.J.

AU - De Waegenaere, A.M.B.

AU - Norde, H.W.

N1 - Pagination: 43

PY - 2012

Y1 - 2012

N2 - Abstract: This paper analyzes risk capital allocation problems. For risk capital allocation problems, the aim is to allocate the risk capital of a firm to its divisions. Risk capital allocation is of central importance in risk-based performance measurement. We consider a case in which the aggregate risk capital is determined via a coherent risk measure. The academic literature advocates an allocation rule that, in game-theoretic terms, is equivalent to using the Aumann-Shapley value as solution concept. This value is however not well-defined in case a differentiability condition is not satisfied. As an alternative, we introduce an allocation rule inspired by the Shapley value in a fuzzy setting. We take a grid on a fuzzy participation set, define paths on this grid and construct an allocation rule based on a path. Then, we define a rule as the limit of the average over these allocations, when the grid size converges to zero. We introduce this rule for a broad class of coherent risk measures. We show that if the Aumann-Shapley value is well-defined, the allocation rule coincides with it. If the Aumann-Shapley value is not defined, which is due to non-differentiability problems, the allocation rule specifies an explicit allocation. It corresponds with the Mertens value, which is originally characterized in an axiomatic way (Mertens, 1988), whereas we provide an asymptotic argument.

AB - Abstract: This paper analyzes risk capital allocation problems. For risk capital allocation problems, the aim is to allocate the risk capital of a firm to its divisions. Risk capital allocation is of central importance in risk-based performance measurement. We consider a case in which the aggregate risk capital is determined via a coherent risk measure. The academic literature advocates an allocation rule that, in game-theoretic terms, is equivalent to using the Aumann-Shapley value as solution concept. This value is however not well-defined in case a differentiability condition is not satisfied. As an alternative, we introduce an allocation rule inspired by the Shapley value in a fuzzy setting. We take a grid on a fuzzy participation set, define paths on this grid and construct an allocation rule based on a path. Then, we define a rule as the limit of the average over these allocations, when the grid size converges to zero. We introduce this rule for a broad class of coherent risk measures. We show that if the Aumann-Shapley value is well-defined, the allocation rule coincides with it. If the Aumann-Shapley value is not defined, which is due to non-differentiability problems, the allocation rule specifies an explicit allocation. It corresponds with the Mertens value, which is originally characterized in an axiomatic way (Mertens, 1988), whereas we provide an asymptotic argument.

KW - capital allocation

KW - risk capital

KW - Aumann-Shapley value

KW - non-differentiability

KW - fuzzy games

M3 - Discussion paper

VL - 2012-091

T3 - CentER Discussion Paper

BT - A Generalization of the Aumann-Shapley Value for Risk Capital Allocation Problems

PB - Econometrics

CY - Tilburg

ER -