254 research outputs found
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Double Chain Ladder
By adding the information of reported count data to a classical triangle of reserving data, we derive a suprisingly simple method for forecasting IBNR and RBNS claims. A simple relationship between development factors allows to involve and then estimate the reporting and payment delay. Bootstrap methods provide prediction errors and make possible the inference about IBNR and RBNS claims, separately
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On the lifetime and one-year views of reserve risk, with application to IFRS 17 and Solvency II risk margins
This paper brings together analytic and simulation-based approaches to reserve risk in general (P&C) insurance, applied to the traditional actuarial view of risk over the lifetime of the liabilities and to the one-year view of Solvency II. It also connects the lifetime and one-year views of risk. The framework of the model in Mack (1993) is used throughout, although the results have wider applicability.
The advantages of a simulation-based approach are highlighted, giving a full predictive distribution, which is used to estimate risk margins under Solvency II and risk adjustments under IFRS 17. We discuss methods for obtaining capital requirements in a cost-of-capital risk margin, and methods for estimating risk adjustments using risk measures applied to a simulated distribution of the outstanding liabilities over their lifetime
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Prediction of RBNS and IBNR claims using claim amounts and claim counts
A model is proposed using the run-off triangle of paid claims and also the numbers of reported claims (in a similar triangular array). These data are usually available, and allow the model proposed to be implemented in a large variety of situations. On the basis of these data, the stochastic model is built from detailed assumptions for individual claims, but then approximated using a compound Poisson framework. The model explicitly takes into account the delay from when a claim is incurred and to when it is reported (the IBNR delay)and the delay from when a claim is reported and to when it is fully paid (the RBNS delay). These two separate sources of delay are estimated separately, unlike most other reserving methods. The results are compared with those of the chain ladder technique
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Continuous Chain Ladder: Reformulating and generalizing a classical insurance problem
The single most important number in the accounts of a non-life insurance company is likely to be the estimate of the outlying liabilities. Since non-life insurance is a major part of our financial industry (amounting to up to 5% of BNP in western countries), it is perhaps surprising that mathematical statisticians and experts of operational research (the natural experts of the underlying problem) have left the intellectual work on estimating this number to actuaries. This paper establishes this important problem in a vocabulary accessible to experts of operations research and mathematical statistics and it can be seen as an open invitation to these two important groups of scholars to join this research. The paper introduces a number of new methodologies and approaches to estimating outstanding liabilities in non-life insurance. In particular it reformulates the classical actuarial technique as a histogram type of approach and improves this classical technique by replacing this histogram by a kernel smoother
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Parameter Reduction in Log-normal Chain-ladder Models
Multiplicative chain-ladder (CL) models are characterized by CL factors that explain the development of claims from one period to the next. In classical CL models every development period has its own CL factor. In the present paper we give a method describing how some of these CL factors can be modeled by a joint functional dependence. This joint functional form reduces the number of model parameters needed
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Validating the double chain ladder stochastic claims reserving model
Double chain ladder, introduced by Martínez-Miranda et al. (2012), is a statistical model to predict outstanding claim reserve. Double chain ladder and Bornhuetter-Ferguson are extensions of the originally described double chain ladder model which gain more stability through including expert knowledge via an incurred claim amounts triangle. In this paper, we introduce a third method, the incurred double chain ladder, which replicates the popular results from the classical chain ladder on incurred data. We will compare and validate these three using two data sets from major property and casualty insurers
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Cash flow simulation for a model of outstanding liabilities based on claim amounts and claim numbers
In this paper we develop a full stochastic cash flow model of outstanding liabilities for the model developed in Verrall, Nielsen and Jessen (2010). This model is based on the simple triangular data available in most non-life insurance companies. By using more data, it is expected that the method will have less volatility than the celebrated chain ladder method. Eventually, our method will lead to lower solvency requirements for those insurance companies that decide to collect counts data and replace their conventional chain ladder method
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Double chain ladder, claims development inflation and zero-claims
Double chain ladder demonstrated how the classical chain ladder technique can be broken down into separate components. It was shown that under certain model assumptions and via one particular estimation technique, it is possible to interpret the classical chain ladder method as a model of the observed number of counts with a built-in delay function from when a claim is reported until it is paid. In this paper, we investigate the double chain ladder model further and consider the case when other knowledge is available, focusing on two specific types of prior knowledge, namely prior knowledge on the number of zero-claims for each underwriting year and prior knowledge about the relationship between the development of the claim and its mean severity. Both types of prior knowledge readily lend themselves to be included in the double chain ladder framework
Estimation of Mortalities
If a linear regression is fit to log-transformed mortalities and the estimate is back-transformed according to the formula Ee^Y = e^{\mu + \sigma^2/2} a systematic bias occurs unless the error distribution is normal and the scale estimate is gauged to normal variance. This result is a consequence of the uniqueness theorem for the Laplace transform.
We determine the systematic bias of minimum-L_2 and minimum-L_1 estimation with sample variance and interquartile range of the residuals as scale estimates under a uniform and four contaminated normal error distributions. Already under innocent looking contaminations the true mortalities may be underestimated by 50% in the long run.
Moreover, the logarithmic transformation introduces an instability into the model that results in a large discrepancy between rg_Huber estimates as the tuning constant regulating the degree of robustness varies.
Contrary to the logarithm the square root stabilizes variance, diminishes the influence of outliers, automatically copes with observed zeros, allows the `nonparametric' back-transformation formula E Y^2 = \mue^2 + \sigma^2, and in the homoskedastic case avoids a systematic bias of minimum-L_2 estimation with sample variance.
For the company-specific table 3 of [Loeb94], in the age range of 20-65 years, we fit a parabola to root mortalities by minimum-L_2 , minimum-L_1, and robust rg_Huber regression estimates, and a cubic and exponential by least squares. The fits thus obtained in the original model are excellent and practically indistinguishable by a \chi^2 goodness-of-fit test.
Finally , dispensing with the transformation of observations, we employ a Poisson generalized linear model and fit an exponential and a cubic by maximum likelihood
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