To split or not to split: Capital allocation with convex risk measures

Convex risk measures were introduced by Deprez and Gerber (1985). Here the problem of allocating risk capital to subportfolios is addressed, when aggregate capital is calculated by a convex risk measure. The Aumann-Shapley 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

Risk measures and economic capital for (re)insurers

This contribution relates to the use of risk measures for determining (re)insurers’ economic capital requirements. Alternative sets of properties of risk measures are discussed. Furthermore, methods for constructing risk measures via indifference arguments, representation results and re-weighting of probability distributions are presented. It is shown how these different approaches relate to popular risk measures, such as VaR, Expected Shortfall, distortion risk measures and the exponential premium principle. The problem of allocating aggregate economic capital to sub-portfolios (e.g. insurers’ lines of business) is then considered, with particular emphasis on marginal-cost-type methods. The relationship between insurance pricing and capital allocation is briefly discussed, based on concepts such as the opportunity and frictional costs of capital and the impact of the potential of default on insurance rates

Modelling: The elephant in the room

In an insurance world preoccupied with Solvency II, internal models have increased in both complexity and business significance. Nonetheless, actuaries are painfully aware of models’ limitations in representing the economic world, not least because of their reliance on often arbitrary assumptions. It is commonplace to say “all models are wrong but some are useful”. But in what sense might an internal model be wrong? If a model is wrong, how can it be useful? Significantly, what incentives does regulation produce for model development and use

High dimensional modelling and simulation with asymmetric normal mixtures

A family of multivariate distributions, based on asymmetric normal mixtures, is introduced in order to model the dependence among insurance and financial risks. The model allows for straight-forward parameterisation via a correlation matrix and enables the modelling of radially asymmetric dependence structures, which are often of interest in risk management applications. Dependence is characterised by showing that increases in correlation values produce models which are ordered in the supermodular order sense. Explicit expressions for the Spearman and Kendall rank correlation coefficients are derived to enable calibration in a copula framework. The model is adapted to simulation in very high dimensions by using Kronecker products, enabling specification of a correlation matrix and an increase in computational spee

Parameter Uncertainty in Exponential Family Tail Estimation

Actuaries are often faced with the task of estimating tails of loss distributions from just a few observations. Thus estimates of tail probabilities (reinsurance prices) and percentiles (solvency capital requirements) are typically subject to substantial parameter uncertainty. We study the bias and MSE of estimators of tail probabilities and percentiles, with focus on 1-parameter exponential families. Using asymptotic arguments it is shown that tail estimates are subject to significant positive bias. Moreover, the use of bootstrap predictive distributions, which has been proposed in the actuarial literature as a way of addressing parameter uncertainty, is seen to double the estimation bias. A bias corrected estimator is thus proposed. It is then shown that the MSE of the MLE, the parametric bootstrap and the bias corrected estimators only differ in terms of order O(n −2), which provides decision-makers with some flexibility as to which estimator to use. The accuracy of asymptotic methods, even for small samples, is demonstrated exactly for the exponential and related distributions, while other 1-parameter distributions are considered in a simulation study. We argue that the presence of positive bias may be desirable in solvency capital calculations, though not necessarily in pricing problems

Model uncertainty in risk capital measurement

The required solvency capital for a financial portfolio is typically given by a tail risk measure such as Value-at-Risk. Estimating the value of that risk measure from a limited, often small, sample of data gives rise to potential errors in the selection of the statistical model and the estimation of its parameters. We propose to quantify the effectiveness of a capital estimation procedure via the notions of residual estimation risk and estimated capital risk. It is shown that for capital estimation procedures that do not require the specification of a model (eg historical simulation) or for worst-case scenario procedures the impact of model uncertainty is substantial, while capital estimation procedures that allow for multiple candidate models using Bayesian methods, partially eliminate model error. In the same setting, we propose a way of quantifying model error that allows to disentangle the impact of model uncertainty from that of parameter uncertainty. We illustrate these ideas by simulation examples considering standard loss and return distributions used in banking and insuranc

Parameter uncertainty and residual estimation risk

The notion of residual estimation risk is introduced to quantify the impact of parameter uncertainty on capital adequacy, for a given risk measure and capital estimation procedure. Residual risk equals the risk measure applied to the difference between a random loss and the corresponding capital estimator. Modified estimation procedures are proposed, based on parametric bootstrapping and predictive distributions, which compensate the impact of parameter uncertainty and lead to higher capital requirements. In the particular case of location-scale families, the analysis simplifies and a capital estimator can always be found that leads to a residual risk of exactly zer