Economics - Research Publications
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ItemAPRA general insurance risk margins( 2006-02)It is common for actuaries to estimate percentile-based risk margins on the assumption of a log normal distribution of liability, together with an estimate of coefficient of variation ("CoV"). This can yield seemingly anomalous results, with percentage risk margin decreasing as CoV increases. The mathematics of this type of risk margin is explored. An APRA risk margin is the maximum of this type and a multiple of CoV. Such risk margins are studied in a more general setting than APRA's, with both percentile p and CoV multiple k free. The APRA risk margins form a special case within this setting.Particular attention is paid to risk margin transition points, values of the log normal dispersion parameter at which the risk margin changes from one form to the other as that parameter increases. For given values of p and k, the existence, uniqueness and location of transition points is investigated. The direction of change of a transition point in the presence of increasing p or k is alsoinvestigated. Various numerical examples are given.
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ItemSecond order Bayesian revision of a generalised linear model( 2005-05)It is well known that the exponential dispersion family (EDF) of univariate distributions is closed under Bayesian revision in the presence of natural conjugate priors. However, this is not the case for the general multivariate EDF. This paper derives a second order approximation to the posterior likelihood of a naturally conjugated generalised linear model (GLM), i.e. multivariate EDF subject to alink function (Section 5.5). It is not the same as a normal approximation. It does, however, lead to second order Bayes estimators of parameters of the posterior.The family of second order approximations is found to be closed under Bayesian revision. This generates a recursion for repeated Bayesian revision of the GLM with theacquisition of additional data. The recursion simplifies greatly for a canonical link. The resulting structure is easily extended to a filter for estimation of the parameters of a dynamic generalised linear model (DGLM) (Section 6.2). The Kalman filter emerges as a special case.A second type of link function, related to the canonical link, and with similar properties, is identified. This is called here the companion canonical link. For a given GLM with canonical link, the companion to that link generates a companion GLM (Section 4). The recursive form of the Bayesian revision of this GLM is also obtained (Section5.5.3). There is a perfect parallel between the development of the GLM recursion and its companion. A dictionary for translation between the two is given so that one is readilyderived from the other (Table 5.1). The companion canonical link also generates a companion DGLM. A filter for this isobtained (Section 6.3).
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ItemLoss reserving with GLMs : a case study( 2004-05)This paper provides a case study in the application of generalised linearmodels (“GLMs”) to loss reserving. The study is motivated by approachingthe exercise from the viewpoint of an actuary with a predisposition to theapplication of the chain ladder (“CL”).The data set under study is seen to violate the conditions for application of theCL in a number of ways. The difficulties of adjusting the CL to allow forthese features of the data are noted (Sections 3).Regression, and particularly GLM regression, is introduced as a structured andrigorous form of data analysis. This enables the investigation and modellingof a number of complex features of the data responsible for the violation of the CL conditions. These include superimposed inflation and changes in the rules governing the payment of claims (Sections 4 to 7).The development of the analysis is traced in some detail, as is the production of a range of diagnostics and tests used to compare candidate models and validate the final one.The benefits of this approach are discussed in Section 8.
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ItemSynchronous bootstrapping of seemingly unrelated regressions( 2005-08)Consider the seemingly unrelated regression framework, in which regression models are applied to a number of data sets, with stochastic dependencies between them. The regression models are not restricted to general linear models (e.g. GLMs). Forecasts are required, with estimates of prediction errors that account for the dependencies between data sets. Bootstrapping is used to estimate prediction errors. Specialised forms of bootstrapping that capture the dependencies are constructed.Insurance and banking applications are mentioned. The former is investigated with numerical examples. The specific context is insurance loss reserving under the requirement that the entire distribution of loss reserve be estimated, where this reserve is aggregated across a number of stochastically dependent lines of business.
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ItemModelling mortgage insurance as a multi-state process( 2006-02)Mortgage insurance claims are considered in Section 2 as an absorbing state in aMarkov chain that involves transitions between the states Healthy, In arrears,Property in Possession, Property sold, Loan discharged, Claim. Section 3considers the representation of this process by a cascade of five frequencyGLMs, and a further GLM for claim size. These models are applied to theforecast of technical liabilities in Section 4, and the estimation of the associatedforecast error in Section 5.
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ItemFurther Observations on Chain Ladder Bias( 2002-01)The chain ladder forecast (CLF) has previously been shown to be biased upward. The present paper calculates the second order approximation to the magnitude of the bias, ie the Taylor series for the bias truncated at terms involving second order moments of observations. Some order relations between data triangles are obtained with respect to this second order bias. While the prediction error of the CLF, as a predictor of loss reserve, does not have zero mean, it does have zero median under certain circumstances. Some numerical consequences are explored.
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ItemChain Ladder Bias( 2002-01)The chain ladder forecast of oustanding losses is known to be unbiased under suitable assumptions. According to these assumptions, claim payments in any cell of a payment triangle are dependent on those from preceding development years of the same accident year. If all cells are assumed stochastically independent, the forecast is no longer unbiased. Section 6 shows that, under very general assumptions, it is biased upward. This result is linked to earlier work on some stochastic versions of the chain ladder.