Capital Allocation \ue0 La Aumann-Shapley for Non Differentiable Risk Measures

We study capital allocation rules satisfying suitable properties for convex and quasi-convex risk measures, by focusing in particular on a family of capital allocation rules based on the dual representation for risk measures and inspired by the Aumann\u2013Shapley allocation principle. These rules extend some well known methods of capital allocation for coherent and convex risk measures to the case of non-Gateauxdifferentiable risk measures. We also analyze the properties of the allocation principles here introduced and discuss their suitability in the quasi-convex context

Capital allocation rules and the no-undercut property

This paper makes the point on a well known property of capital allocation rules, namely the one called no-undercut. Its desirability in capital allocation stems from some stability game theoretical features related to the notion of core, both for finite and infinite games. We review these aspects, by relating them to the properties of the risk measures involved in capital allocation problems. We also discuss some problems and possible extensions arising when we deal with non-coherent risk measures

The asymptotic nucleolus of large monopolistic games

We study the asymptotic nucleolus of large differentiable monopolistic games. We show that if v is a monopolistic game which is a composition of a non-decreasing concave and differentiable function with a vector of measures, then v has an asymptotic nucleolus. We also provide an explicit formula for the asymptotic nucleolus of v and show that it coincides with the center of symmetry of the subset of the core of v in which all the monopolists obtain the same payoff. We apply this result to large monopolistic market games to obtain a relationship between the asymptotic nucleolus of the game and the competitive payoff distributions of the market

Properties of risk capital allocation methods: Core Compatibility, Equal Treatment Property and Strong Monotonicity

In finance risk capital allocation raises important questions both from theoretical and practical points of view. How to share risk of a portfolio among its subportfolios? How to reserve capital in order to hedge existing risk and how to assign this to different business units? We use an axiomatic approach to examine risk capital allocation, that is we call for fundamental properties of the methods. Our starting point is Csóka and Pintér (2011) who show by generalizing Young (1985)'s axiomatization of the Shapley value that the requirements of Core Compatibility, Equal Treatment Property and Strong Monotonicity are irreconcilable given that risk is quantified by a coherent measure of risk. In this paper we look at these requirements using analytic and simulations tools. We examine allocation methods used in practice and also ones which are theoretically interesting. Our main result is that the problem raised by Csóka and Pintér (2011) is indeed relevant in practical applications, that is it is not only a theoretical problem. We also believe that through the characterizations of the examined methods our paper can serve as a useful guide for practitioners

Evaluating Capital Allocation Below Portfolio Level

This thesis explores the ability for retail banks to allocate economic capital below portfolio level. First, a discussion about capital requirements and risk measures to provide a sound basis for determining the economic capital of the bank. In general, economic capital is allocated to the banks portfolios but not on a more granular level, through a capital allocation method. This study discuss three dierent approaches for allocation of economic capital below portfolio level; game theory, nance and optimization. Both the game theory and nance approach reach the same conclusion, that the best allocation principle is the gradient of the risk measure. The optimization method allocates economic capital through minimization of a concept called risk residual, which conclude that the optimal allocation is derived from the marginal distribution of a customer. Capital allocation below portfolio level give the management a good overview of risks from dierent customers. In order to determine the performance of the portfolios in the bank a Risk-Adjusted-Return-On-Capital is used, with economic capital as input. The thesis include some comments about how the choice of capital allocation methods aect the performance measurement. The thesis concludes with an evaluation of the methods by simulations of a ctional bank conducted in the software R. Key Words: Risk Appetite, Economic Capital, Risk measure, Capital Allocation Methods, Allocation Below Portfolio level, Game theory, Optimization, Marginal Contributio

A risk management approach to capital allocation

The European insurance sector will soon be faced with the application of Solvency 2 regulation norms. It will create a real change in risk management practices. The ORSA approach of the second pillar makes the capital allocation an important exercise for all insurers and specially for groups. Considering multi-branches firms, capital allocation has to be based on a multivariate risk modeling. Several allocation methods are present in the literature and insurers practices. In this paper, we present a new risk allocation method, we study its coherence using an axiomatic approach, and we try to define what the best allocation choice for an insurance group is

Risk contributions of lambda quantiles

Risk contributions of portfolios form an indispensable part of risk adjusted performance measurement. The risk contribution of a portfolio, e.g., in the Euler or Aumann-Shapley framework, is given by the partial derivatives of a risk measure applied to the portfolio return in direction of the asset weights. For risk measures that are not positively homogeneous of degree 1, however, known capital allocation principles do not apply. We study the class of lambda quantile risk measures, that includes the well-known Value-at-Risk as a special case, but for which no known allocation rule is applicable. We prove differentiability and derive explicit formulae of the derivatives of lambda quantiles with respect to their portfolio composition, that is their risk contribution. For this purpose, we define lambda quantiles on the space of portfolio compositions and consider generic (also non-linear) portfolio operators. We further derive the Euler decomposition of lambda quantiles for generic portfolios and show that lambda quantiles are homogeneous in the space of portfolio compositions, with a homogeneity degree that depends on the portfolio composition and the lambda function. This result is in stark contrast to the positive homogeneity properties of risk measures defined on the space of random variables which admit a constant homogeneity degree. We introduce a generalised version of Euler contributions and Euler allocation rule, which are compatible with risk measures of any homogeneity degree and non-linear portfolios. We further provide financial interpretations of the homogeneity degree of lambda quantiles and introduce the notion of event-specific homogeneity of portfolio operators

Methods of capital allocation in a Solvency II environment

Mestrado em Actuarial ScienceDe acordo com a regulamentação de Solvência II, o SCR é geralmente calculado usando uma fórmula padrão que considera os riscos que uma seguradora enfrenta. Devido à agregação dos diferentes riscos, são originados benefícios de diversificação e um valor de SCR total menor que a soma dos requisitos de capital de cada risco. Para ter em conta estes benefícios de diversificação, o capital total deve ser alocado de volta aos níveis mais baixos de risco, aplicando um método apropriado de alocação de capital. Este relatório é resultado de um estágio curricular que decorreu na EY. Um dos objetivos foi encontrar o método mais apropriado para realizar a alocação do SCR de uma empresa de seguros. Foram estudados cinco métodos de alocação, Proporcional, Variância-Covariância, Merton e Perold, Shapley e Euler. Os métodos são comparados teoricamente, analisando as suas respetivas propriedades e, com base em vários estudos presentes na literatura, conclui-se que o método de Euler é o mais apropriado. Este trabalho contribui para uma melhor compreensão dos métodos de alocação de capital e permite demonstrar como alocar o SCR. Contribui também para mostrar como construir o SES para fins do cálculo do ajustamento LAC DT. Visto que esta tarefa foi uma das dificuldades referidas no QIS 5, este trabalho pode servir como base literária, sendo útil para superar essas dificuldades.Under Solvency II regulation the SCR is mainly calculated using a standard formula which considers the risks that an insurer faces. Due to this aggregation of risks, a diversification benefit is achieved and the global SCR is smaller than the sum of the capital requirements of each risk. To take these diversification benefits into account the total capital should be allocated back to the lower levels of risk by applying a proper method of capital allocation. This report is the result of a curricular internship that took place at EY. One of the goals was to find the most appropriate method to perform a capital allocation of the SCR of an insurance company. Five methods of allocation were studied, Proportional, Variance-Covariance, Merton and Perold, Shapley and Euler. The methods were compared theoretically by analyzing their respective properties, and based on several studies in the literature it is concluded that the Euler method is the most appropriate to apply. This report contributes to a better understanding of capital allocation methods and allows to demonstrate how to allocate the SCR. It also contributes to show how to construct the SES for the purpose of the calculation of the adjustment of LAC DT. Since this task was one of the difficulties enumerated in the Fifth Quantitative Impact Study (QIS 5), this work can serve as a literary base, being useful to overcome these difficulties.info:eu-repo/semantics/publishedVersio