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ItemHow to proxy the unmodellable: Analysing granular insurance claims in the presence of unobservable or complex driversAvanzi, B ; Taylor, G ; wong, B ; Xian, A (Institute of Actuaries, Australia, 2018)The estimation of claim and premium liabilities is a key component of an actuary's role and plays a vital part of any insurance company’s operations. In practice, such calculations are complicated by the stochastic nature of the claims process as well as the impracticality of capturing all relevant and material drivers of the observed claims data. In the past, computational limitations have promoted the prevalence of simplified (but possibly suboptimal) aggregate methodologies. However, in light of modern advances in processing power, it is viable to increase the granularity at which we analyse insurance data sets so that potentially useful information is not discarded. By utilising more granular and detailed data (that is usually readily available to insurers), model predictions may become more accurate and precise. Unfortunately, detailed analysis of large insurance data sets in this manner poses some unique challenges. Firstly, there is no standard framework to which practitioners can refer and it can be challenging to tractably integrate all modelled components into one comprehensive model. Secondly, analysis at greater granularity or level of detail requires more intense levels of scrutiny as complex trends and drivers that were previously masked by aggregation and discretisation assumptions may emerge. This is particularly an issue with claim drivers that are either unobservable to the modeller or very difficult/expensive to model. Finally, computation times are a material concern when processing such large volumes of data as model outputs need to be obtained in reasonable timeframes. Our proposed methodology overcomes the above problems by using a Markovmodulated nonhomogeneous Poisson process framework. This extends the standard Poisson model by allowing for overdispersion to be captured in an interpretable, structural manner. The approach implements a flexible exposure measure to explicitly allow for known/modelled claim drivers while the hidden component of the Hidden Markov model captures the impact of unobservable or practicably nonmodellable information. Computational developments are made to drastically reduce calibration times. Theoretical findings are illustrated and validated in an empirical case study using Australian general insurance data in order to highlight the benefits of the proposed approach.

ItemOn the Impact, Detection and Treatment of Outliers in Robust Loss ReservingAvanzi, B ; Taylor, G ; Wong, B ; Lavendar, M (Actuaries Institute, 2016)The sensitivity of loss reserving techniques to outliers in the data or deviations from model assumptions is a well known challenge. For instance, it has been shown that the popular chainladder reserving approach is at significant risk to such aberrant observations in that reserve estimates can be significantly shifted in the presence of even one outlier. In this paper we firstly investigate the sensitivity of reserves and mean squared errors of prediction under Mack's Model. This is done through the derivation of impact functions which are calculated by taking the first derivative of the relevant statistic of interest with respect to an observation. We also provide and discuss the impact functions for quantiles when total reserves are assumed to be lognormally distributed. Additionally, comparisons are made between the impact functions for individual accident year reserves under Mack's Model and the BornhuetterFerguson methodology. It is shown that the impact of incremental claims on these statistics of interest varies widely throughout a loss triangle and is heavily dependent on other cells in the triangle. We then put forward two alternative robust bivariate chainladder techniques (Verdonck and VanWouwe, 2011) based on AdjustedOutlyingness (Hubert and Van der Veeken, 2008) and bagdistance (Hubert et al., 2016). These techniques provide a measure of outlyingness that is unique to each individual observation rather than largely relying on graphical representations as is done under the existing bagplot methodology. Furthermore the Adjusted Outlyingness approach explicitly incorporates a robust measure of skewness into the analysis whereas the bagplot captures the shape of the data only through a measure of rank. Results are illustrated on two sets of real bivariate data from general insurers.

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.