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ItemCapturing nonexchangeable dependence in multivariate loss processes with nested Archimedean Levy copulasAvanzi, B ; Tao, J ; Wong, B ; Yang, X (Cambridge University Press (CUP), 20160301)The class of spectrally positive Lévy processes is a frequent choice for modelling loss processes in areas such as insurance or operational risk. Dependence between such processes (e.g. between different lines of business) can be modelled with Lévy copulas. This approach is a parsimonious, efficient and flexible method which provides many of the advantages akin to distributional copulas for random variables. Literature on Lévy copulas seems to have primarily focussed on bivariate processes. When multivariate settings are considered, these usually exhibit an exchangeable dependence structure (whereby all subset of the processes have an identical marginal Lévy copula). In reality, losses are not always associated in an identical way, and models allowing for nonexchangeable dependence patterns are needed. In this paper, we present an approach which enables the development of such models. Inspired by ideas and techniques from the distributional copula literature we investigate the procedure of nesting Archimedean Lévy copulas. We provide a detailed analysis of this construction, and derive conditions under which valid multivariate (nested) Lévy copulas are obtained. Our results are discussed and illustrated, notably with an example of model fitting to data.

ItemCORRELATIONS BETWEEN INSURANCE LINES OF BUSINESS: AN ILLUSION OR A REAL PHENOMENON? SOME METHODOLOGICAL CONSIDERATIONSAvanzi, B ; Taylor, G ; Wong, B (Cambridge University Press (CUP), 20160501)This paper is concerned with dependency between business segments in the nonlife insurance industry. When considering the business of an insurance company at the aggregate level, dependence structures can have a major impact in several areas of Enterprise Risk Management, such as in claims reserving and capital modelling. The accurate estimation of the diversification benefits related to the dependence structures between lines of business (LoBs) is crucial for (i) capital efficiency, as one should avoid holding unnecessarily high levels of capital, and (ii) solvency of the insurance company, as an underestimation, on the other hand, may lead to insufficient capitalisation and safety. There seems to be a great deal of preconception as to how dependent insurance claims should be. Often, presence of dependence is taken as a given and rarely discussed or challenged, perhaps because of the lack of extensive datasets to be publicly analysed. In this paper, we take a different approach, and consider how much correlation some real datasets actually display (the Meyers–Shi dataset from the USA, and the AUSI dataset from Australia). We develop a simple theoretical framework that enables us to explain how and why correlations can be illusory (and what we mean by that). We show with some real examples that, sometimes, most (if not all) of the correlation can be “explained” by an appropriate methodology. Two major conclusions stem from our analysis. 1.In any attempt to measure crossLoB correlations, careful modelling of the data needs to be the order of the day. The exercise will not be well served by rough modelling, such as the use of simple chain ladders, and may indeed result in the prescription of excessive risk margins and/or capital margins. 2.Such empirical evidence as examined in the paper reveals crossLoB correlations that vary only in the range zero to very modest. There is little evidence in favour of the high correlation assumed in some jurisdictions. The evidence suggests that these assumptions derived from either poor modelling or a misconception of the crossLoB dependencies relevant to the purpose to which they are applied.

ItemOn the Interface Between Optimal Periodic and Continuous Dividend Strategies in the Presence of Transaction CostsAvanzi, B ; Tu, V ; Wong, B (Cambridge University Press (CUP), 20160901)In the classical optimal dividends problem, dividend decisions are allowed to be made at any point in time — according to a continuous strategy. Depending on the surplus process that is considered and whether dividend payouts are bounded or not, optimal strategies are generally of a band, barrier or threshold type. In reality, while surpluses change continuously, dividends are generally paid on a periodic basis. Because of this, the actuarial literature has recently considered strategies where dividends are only allowed to be distributed at (random) discrete times — according to a periodic strategy. In this paper, we focus on the Brownian risk model. In this context, the optimal continuous and periodic strategies have previously been shown (independently of one another) to be of barrier type. For the first time, we consider a model where both strategies are used. In such a hybrid strategy, decisions are allowed to be made either at any time (continuously), or periodically at a lower cost. This proves optimal in some cases. We also determine under which combination of parameters a pure continuous, pure periodic or hybrid (including both continuous and periodic dividend payments) barrier strategy is optimal. Interestingly, the hybrid strategy lies inbetween periodic and continuous strategies, which provides some interesting insights. Results are illustrated.

ItemA microlevel claim count model with overdispersion and reporting delaysAvanzi, B ; Wong, B ; Yang, X (Elsevier, 20161101)The accurate estimation of outstanding liabilities of an insurance company is an essential task. This is to meet regulatory requirements, but also to achieve efficient internal capital management. Over the recent years, there has been increasing interest in the utilisation of insurance data at a more granular level, and to model claims using stochastic processes. So far, this socalled ‘microlevel reserving’ approach has mainly focused on the Poisson process. In this paper, we propose and apply a Cox process approach to model the arrival process and reporting pattern of insurance claims. This allows for overdispersion and serial dependency in claim counts, which are typical features in real data. We explicitly consider risk exposure and reporting delays, and show how to use our model to predict the numbers of IncurredButNotReported (IBNR) claims. The model is calibrated and illustrated using real data from the AUSI data set.

ItemStochastic loss reserving with dependence: A flexible multivariate Tweedie approachAvanzi, B ; Taylor, G ; Phuong, AV ; Wong, B (Elsevier, 20161101)Stochastic loss reserving with dependence has received increased attention in the last decade. A number of parametric multivariate approaches have been developed to capture dependence between lines of business within an insurer’s portfolio. Motivated by the richness of the Tweedie family of distributions, we propose a multivariate Tweedie approach to capture cellwise dependence in loss reserving. This approach provides a transparent introduction of dependence through a common shock structure. In addition, it also has a number of ideal properties, including marginal flexibility, transparency, and tractability including moments that can be obtained in closed form. Theoretical results are illustrated using both simulated data sets and a real data set from a propertycasualty insurer in the US.

ItemA Note on Realistic Dividends in Actuarial Surplus ModelsAvanzi, B ; Tu, V ; Wong, B (MDPI, 20161201)Because of the profitable nature of risk businesses in the long term, de Finetti suggested that surplus models should allow for cash leakages, as otherwise the surplus would unrealistically grow (on average) to infinity. These leakages were interpreted as ‘dividends’. Subsequent literature on actuarial surplus models with dividend distribution has mainly focussed on dividend strategies that either maximise the expected present value of dividends until ruin or lead to a probability of ruin that is less than one (see Albrecher and Thonhauser, Avanzi for reviews). An increasing number of papers are directly interested in modelling dividend policies that are consistent with actual practice in financial markets. In this short note, we review the corporate finance literature with the specific aim of fleshing out properties that dividend strategies should ideally satisfy, if one wants to model behaviour that is consistent with practice.

ItemOn the Distribution of the Excedents of Funds with Assets and Liabilities in Presence of Solvency and Recovery RequirementsAvanzi, B ; Henriksen, LFB ; Wong, B (Cambridge University Press (CUP), 20180501)We consider a profitable, risky setting with two separate, correlated asset and liability processes (first introduced by Gerber and Shiu, 2003). The company that is considered is allowed to distribute excess profits (traditionally referred to as dividends in the literature), but is regulated and is subject to particular regulatory (solvency) constraints. Because of the bivariate nature of the surplus formulation, such distributions of excess profits can take two alternative forms. These can originate from a reduction of assets (and hence a payment to owners), but also from an increase of liabilities (when these represent the wealth of owners, such as in pension funds). The latter is particularly relevant if distributions of assets do not make sense because of the context, such as in regulated pension funds where assets are locked until retirement. In this paper, we extend the model of Gerber and Shiu (2003) and consider recovery requirements for the distribution of excess funds. Such recovery requirements are an extension of the plain vanilla solvency constraints considered in Paulsen (2003), and require funds to reach a higher level of funding than the solvency level (if and after it is triggered) before excess funds can be distributed again. We obtain closedform expressions for the expected present value of distributions (asset decrements or liability increments) when a distribution barrier is used.

ItemOn optimal joint reflective and refractive dividend strategies in spectrally positive Levy modelsAvanzi, B ; Perez, JL ; Wong, B ; Yamazaki, K (Elsevier, 20170101)The expected present value of dividends is one of the classical stability criteria in actuarial risk theory. In this context, numerous papers considered threshold (refractive) and barrier (reflective) dividend strategies. These were shown to be optimal in a number of different contexts for bounded and unbounded payout rates, respectively. In this paper, motivated by the behavior of some dividend paying stock exchange companies, we determine the optimal dividend strategy when both continuous (refractive) and lump sum (reflective) dividends can be paid at any time, and if they are subject to different transaction rates. We consider the general family of spectrally positive Lévy processes. Using scale functions, we obtain explicit formulas for the expected present value of dividends until ruin, with a penalty at ruin. We develop a verification lemma, and show that a twolayer strategy is optimal. Such a strategy pays continuous dividends when the surplus exceeds level , and all of the excess over as lump sum dividend payments. Results are illustrated.

ItemOptimal dividends under Erlang(2) interdividend decision timesAvanzi, B ; Tu, V ; Wong, B (Elsevier, 20180301)In the classical dividends problem, dividend decisions are allowed to be made at any time. Under such a framework, the optimal dividend strategies are often of barrier or threshold type, which can lead to very irregular dividend payments over time. In practice however companies distribute dividends on a periodic basis. In that spirit, “Erlangisation” techniques have been used to approximate problems with fixed interdividend decision times. When studying the optimality of such strategies, the existing literature focuses exclusively on the special case of exponential – that is, Erlang(1) – interdividend decision times. Higher dimensional models are surprisingly difficult to study due to the implicit nature of some of the equations. While some of this difficulty continues to exist in high dimensions, in this paper we provide a stepping stone to the general Erlang() problem by providing a detailed analysis of the optimality of periodic barrier strategies when interdividenddecision times are Erlang(2) distributed. Results are illustrated.

ItemCOMMON SHOCK MODELS FOR CLAIM ARRAYSAvanzi, B ; Taylor, G ; Wong, B (Cambridge University Press (CUP), 20180901)The paper is concerned with multiple claim arrays. In recognition of the extensive use by practitioners of large correlation matrices for the estimation of diversification benefits in capital modelling, we develop a methodology for the construction of such correlation structures (to any dimension). Indeed, the literature does not document any methodology by which practitioners, who often parameterise those correlations by means of informed guesswork, may do so in a disciplined and parsimonious manner. We construct a broad and flexible family of models, where dependency is induced by common shock components. Models incorporate dependencies between observations both within arrays and between arrays. Arrays are of general shape (possibly with holes), but include the usual cases of claim triangles and trapezia that appear in the literature. General forms of dependency are considered with cell, row, column, diagonalwise, and other forms of dependency as special cases. Substantial effort is applied to practical interpretation of such matrices generated by the models constructed here. Reasonably realistic examples are examined, in which an expression is obtained for the general entry in the correlation matrix in terms of a limited set of parameters, each of which has a straightforward intuitive meaning to the practitioner. This will maximise chance of obtaining a reliable matrix. This construction is illustrated by a numerical example.