51 research outputs found
Optimal capital allocation principles.
This paper develops a unifying framework for allocating the aggregate capital of a financial firm to its business units. The approach relies on an optimisation argument, requiring that the weighted sum of measures for the deviations of the business unitâs losses from their respective allocated capitals be minimised. This enables the association of alternative allocation rules to specific decision criteria and thus provides the risk manager with flexibility to meet specific target objectives. The underlying general framework reproduces many capital allocation methods that have appeared in the literature and allows for several possible extensions. An application to an insurance market with policyholder protection is additionally provided as an illustration.
Optimal capital allocation principles
This paper develops a unifying framework for allocating the aggregate capital of a financial firm to its business units. The approach relies on an optimisation argument, requiring that the weighted sum of measures for the deviations of the business unitâs losses from their respective allocated capitals be minimised. This enables the association of alternative allocation rules to specific decision criteria and thus provides the risk manager with flexibility to meet specific target objectives. The underlying general framework reproduces many capital allocation methods that have appeared in the literature and allows for several possible extensions. An application to an insurance market with policyholder protection is additionally provided as an illustration.Capital allocation; risk measure; comonotonicity; Euler allocation; default option; Lloydâs of London
Risk exchange with distorted probabilities Topic 2: Risk finance and risk transfer
Abstract We study the equilibrium in a risk exchange, where agents' preferences are characterised by generalised (rank-dependent) expected utility, i.e. by a concave utility and a convex probability distortio
Risk margin for a non-life insurance run-off
For solvency purposes insurance companies need to calculate so-called best-estimate reserves for outstanding loss liability cash flows and a corresponding risk margin for non-hedgeable insurance-technical risks in these cash flows. In actuarial practice, the calculation of the risk margin is often not based on a sound model but various simplified methods are used. In the present paper we properly define these notions and we introduce insurance-technical probability distortions. We describe how the latter can be used to calculate a risk margin for non-life insurance run-off liabilities in a mathematically consistent wa
Differential Sensitivity in Discontinuous Models
Differential sensitivity measures provide valuable tools for interpreting
complex computational models used in applications ranging from simulation to
algorithmic prediction. Taking the derivative of the model output in direction
of a model parameter can reveal input-output relations and the relative
importance of model parameters and input variables. Nonetheless, it is unclear
how such derivatives should be taken when the model function has
discontinuities and/or input variables are discrete. We present a general
framework for addressing such problems, considering derivatives of
quantile-based output risk measures, with respect to distortions to random
input variables (risk factors), which impact the model output through
step-functions. We prove that, subject to weak technical conditions, the
derivatives are well-defined and derive the corresponding formulas. We apply
our results to the sensitivity analysis of compound risk models and to a
numerical study of reinsurance credit risk in a multi-line insurance portfolio
A Discussion of Discrimination and Fairness in Insurance Pricing
Indirect discrimination is an issue of major concern in algorithmic models.
This is particularly the case in insurance pricing where protected policyholder
characteristics are not allowed to be used for insurance pricing. Simply
disregarding protected policyholder information is not an appropriate solution
because this still allows for the possibility of inferring the protected
characteristics from the non-protected ones. This leads to so-called proxy or
indirect discrimination. Though proxy discrimination is qualitatively different
from the group fairness concepts in machine learning, these group fairness
concepts are proposed to 'smooth out' the impact of protected characteristics
in the calculation of insurance prices. The purpose of this note is to share
some thoughts about group fairness concepts in the light of insurance pricing
and to discuss their implications. We present a statistical model that is free
of proxy discrimination, thus, unproblematic from an insurance pricing point of
view. However, we find that the canonical price in this statistical model does
not satisfy any of the three most popular group fairness axioms. This seems
puzzling and we welcome feedback on our example and on the usefulness of these
group fairness axioms for non-discriminatory insurance pricing.Comment: 14 page
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Optimal capital allocation in a hierarchical corporate structure
We consider capital allocation in a hierarchical corporate structure where stakeholders at two organizational levels (e.g., board members vs line managers) may have conflicting objectives, preferences, and beliefs about risk. Capital allocation is considered as the solution to an optimization problem whereby a quadratic deviation measure between individual losses (at both levels) and allocated capital amounts is minimized. Thus, this paper generalizes the framework of Dhaene et al. (2012), by allowing potentially diverging risk preferences in a hierarchical structure. An explicit unique solution to this optimization problem is given. In several examples, it is shown how the optimal capital allocation achieves a compromise between conflicting views of risk within the organization
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Risk measurement in the presence of background risk
A distortion-type risk measure is constructed, which evaluates the risk of any uncertain position in the context of a portfolio that contains that position and a fixed background risk. The risk measure can also be used to assess the performance of individual risks within a portfolio, allowing for the portfolioâs re-balancing, an area where standard capital allocation methods fail. It is shown that the properties of the risk measure depart from those of coherent distortion measures. In particular, it is shown that the presence of background risk makes risk measurement sensitive to the scale and aggregation of risk. The case of risks following elliptical distributions is examined in more detail and precise characterisations of the risk measureâs aggregation properties are obtained
Optimal capital allocation principles
This paper develops a unifying framework for allocating the aggregate capital of a financial firm to its business units. The approach relies on an optimisation argument, requiring that the weighted sum of measures for the deviations of the business unitâs losses from their respective allocated capitals be minimised. This enables the association of alternative allocation rules to specific decision criteria and thus provides the risk manager with flexibility to meet specific target objectives. The underlying general framework reproduces many capital allocation methods that have appeared in the literature and allows for several possible extensions. An application to an insurance market with policyholder protection is additionally provided as an illustration
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To split or not to split: capital allocation with convex risk measures
Convex risk measures were introduced by Deprez and Gerber [Deprez, O., Gerber, H.U., 1985. On convex principles of premium calculation. Insurance: Math. Econom. 4 (3), 179â189]. Here the problem of allocating risk capital to subportfolios is addressed, when convex risk measures are used. The AumannâShapley value is proposed as an appropriate allocation mechanism. Distortion-exponential measures are discussed extensively and explicit capital allocation formulas are obtained for the case that the risk measure belongs to this family. Finally the implications of capital allocation with a convex risk measure for the stability of portfolios are discussed. It is demonstrated that using a convex risk measure for capital allocation can produce an incentive for infinite fragmentation of portfolios
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