Diagonal effects in claims reserving

In this paper we present two different approaches to how one can include diagonal effects in non-life claims reserving based on run-off triangles. Empirical analyses suggest that the approaches in Zehnwirth (2003) and Kuang et al. (2008a, 2008b) do not work well with low-dimensional run-off triangles because estimation uncertainty is too large. To overcome this problem we consider similar models with a smaller number of parameters. These are closely related to the framework considered in Verbeek (1972) and Taylor (1977, 2000); the separation method. We explain that these models can be interpreted as extensions of the multiplicative Poisson models introduced by Hachemeister & Stanard (1975) and Mack (1991).     

“The Future of Risk Classification in the Age of Predictive DNA-Based Testing”, Donald C. Chambers,January 1999

Abstract

The correlation among multiple lines of business plays an important role in quantifying the uncertainty of loss reserves for insurance portfolios. To accommodate correlation, most multivariate loss-reserving methods focus on the pairwise association between corresponding cells in multiple run-off triangles. However, such practice usually relies on the independence assumption across accident years and ignores the calendar year effects that could affect all open claims simultaneously and induce dependencies among loss triangles. To address this issue, we study a Bayesian log-normal model in the prediction of outstanding claims for dependent lines of business. In addition to the pairwise correlation, our method allows for an explicit examination of the correlation due to common calendar year effects. Further, different specifications of the calendar year trend are considered to reflect valuation actuaries’ prior knowledge of claim development. In a case study, we analyze an insurance portfolio of personal and commercial auto lines from a major U.S. property-casualty insurer. It is shown that the incorporation of calendar year effects improves model fit significantly, though it contributes substantively to the predictive variability. The availability of the realizations of predicted claims permits us to perform a retrospective test, which suggests that extra prediction uncertainty is indispensable in modern risk management practices.     

Prognose des Abwicklungsergebnisses mittels Chain-Ladder-Verfahren für abhängige Run-Off-Portfolios

In the present paper we analyse how the estimators from Merz u. Wüthrich (2007) could be generalised to the case of N correlated run-off triangles. The simultaneous view on N correlated subportfolios is motivated by the fact, that in practice a run-off portfolio often has to be divided in subportfolios, so that the homogeneity assumption of the claims reserving method on each subportfolio is satisfied. We derive an explicit formula for the process-variance, the estimation-error and the prediction error made by the forecast for the claims development result with the Chain-Ladder method. We illustrate the results by an example.     

A Robustification of the Chain-Ladder Method

Abstract

In a non-life insurance business an insurer often needs to build up a reserve to able to meet his or her future obligations arising from incurred but not reported completely claims. To forecast these claims reserves, a simple but generally accepted algorithm is the classical chain-ladder method. Recent research essentially focused on the underlying model for the claims reserves to come to appropriate bounds for the estimates of future claims reserves. Our research concentrates on scenarios with outlying data. On closer examination it is demonstrated that the forecasts for future claims reserves are very dependent on outlying observations. The paper focuses on two approaches to robustify the chain-ladder method: the first method detects and adjusts the outlying values, whereas the second method is based on a robust generalized linear model technique. In this way insurers will be able to find a reserve that is similar to the reserve they would have found if the data contained no outliers. Because the robust method flags the outliers, it is possible to examine these observations for further examination. For obtaining the corresponding standard errors the bootstrapping technique is applied. The robust chain-ladder method is applied to several run-off triangles with and without outliers, showing its excellent performance.     

Unallocated loss adjustment expense reserving

In non-life insurance, the provision for outstanding claims (the claims reserve) should include future loss adjustment expenses, i.e. administrative expenses to settle the claims, and therefore we have to estimate the expected Unallocated Loss Adjustment Expenses (ULAE) – expenses that are not attributable to individual claims, such as salaries at the claims handling department. The ULAE reserve has received little attention from European actuaries in the literature, supposedly because of the lack of detailed data for estimation and evaluation. Having good estimation procedures will, however, become even more important with the introduction of the Solvency II regulations, which require unbiased estimation of future cash flows for all expenses. We present a model for ULAE at the individual claim level that includes both fixed and variable costs. This model leads to an estimate of the ULAE reserve at the aggregate (line-of-business) level, as demonstrated in a numerical example from a Swedish non-life insurer.     

Mittlerer quadratischer Prognosefehler bei der Prognose des Abwicklungsergebnisses mittels der Chain-Ladder-Methode

Often in non-life insurance claims reserves are the largest position on the liability side of the balance sheet. The determination of adequate claims reserves in two consecutive accounting years leads to the so-called development result, which is defined as the difference of two successive predictors for the claims reserves. If the predictors for the claims reserves are unbiased the expected development result is equal to zero. However, since in claims reserving one predicts future payments the observed development result will in general deviate from the expected value. In the present paper we analyze this deviation. In an example we discuss the results.     

Das stochastische Re-Reserving

Next to premium risk, reserve risk is an essential part of underwriting risk for property and casualty insurers. Up to now, the literature on non-life reserve risk for the most part only considered an ultimative point of view, which means that the reserve risk uncertainty would be quantified up to final settlement. In the context of the new Solvency II project, as well as against the background of internal models in a several year context, for now an intensified discussion about a one-year horizon has occurred. Thereby, the reserve risk uncertainty would only be quantified up to a one-year claims development result. In this paper we would like to present a simulation method, called stochastic Re-Reserving for modelling the reserve risk on a yearly basis. In addition to that, we would like to make a paradigmatic comparison between stochastic Re-Reserving and an analytical approach, called claims development result method (see Merz and Wuethrich, Modelling the claims development result for solvency purposes. In: CAS E-Forum, pp. 542–568, 2008).     

Micro-level stochastic loss reserving for general insurance

The vast literature on stochastic loss reserving concentrates on data aggregated in run-off triangles. However, a triangle is a summary of an underlying data-set with the development of individual claims. We refer to this data-set as ‘micro-level’ data. Using the framework of Position Dependent Marked Poisson Processes) and statistical tools for recurrent events, a data-set is analyzed with liability claims from a European insurance company. We use detailed information of the time of occurrence of the claim, the delay between occurrence and reporting to the insurance company, the occurrences of payments and their sizes, and the final settlement. Our specifications are (semi)parametric and our approach is likelihood based. We calibrate our model to historical data and use it to project the future development of open claims. An out-of-sample prediction exercise shows that we obtain detailed and valuable reserve calculations. For the case study developed in this paper, the micro-level model outperforms the results obtained with traditional loss reserving methods for aggregate data.     

Robust bootstrap procedures for the chain-ladder method

Insurers are faced with the challenge of estimating the future reserves needed to handle historic and outstanding claims that are not fully settled. A well-known and widely used technique is the chain-ladder method, which is a deterministic algorithm. To include a stochastic component one may apply generalized linear models to the run-off triangles based on past claims data. Analytical expressions for the standard deviation of the resulting reserve estimates are typically difficult to derive. A popular alternative approach to obtain inference is to use the bootstrap technique. However, the standard procedures are very sensitive to the possible presence of outliers. These atypical observations, deviating from the pattern of the majority of the data, may both inflate or deflate traditional reserve estimates and corresponding inference such as their standard errors. Even when paired with a robust chain-ladder method, classical bootstrap inference may break down. Therefore, we discuss and implement several robust bootstrap procedures in the claims reserving framework and we investigate and compare their performance on both simulated and real data. We also illustrate their use for obtaining the distribution of one year risk measures.     

Spatial modelling of claim frequency and claim size in non-life insurance

In this paper, models for claim frequency and average claim size in non-life insurance are considered. Both covariates and spatial random effects are included allowing the modelling of a spatial dependency pattern. We assume a Poisson model for the number of claims, while claim size is modelled using a Gamma distribution. However, in contrast to the usual compound Poisson model, we allow for dependencies between claim size and claim frequency. A fully Bayesian approach is followed, parameters are estimated using Markov Chain Monte Carlo (MCMC). The issue of model comparison is thoroughly addressed. Besides the deviance information criterion and the predictive model choice criterion, we suggest the use of proper scoring rules based on the posterior predictive distribution for comparing models. We give an application to a comprehensive data set from a German car insurance company. The inclusion of spatial effects significantly improves the models for both claim frequency and claim size, and also leads to more accurate predictions of the total claim sizes. Further, we detect significant dependencies between the number of claims and claim size. Both spatial and number of claims effects are interpreted and quantified from an actuarial point of view.     

Robustifizierung des Chain-Ladder-Verfahrens über ein skalares Zustandsraummodell

Applications of state space models and the Kalman filter are comparatively underrepresented in stochastic claims reserving. This is usually caused by their high complexity due to matrix-based approaches, which complicate their applications. In order to facilitate the implementation of state space models in practice, we present a state space model for cumulative payments in the framework of a scalar-based approach. In addition to a comprehensive presentation of this scalar state space model, some empirical applications and comparisons with popular stochastic claims reserving methods are performed, which show the strengths of the scalar state space model in practical applications. This model is a robustified extension of the well-known Chain Ladder method under the assumption, that the observations in the upper triangle are based on unobservable states. Using Kalman-filter recursions for prediction, filtering and smoothing of cumulative payments, the entire unobservable lower and upper run-off triangles can be determined. Moreover, the model provides an easy way to find and smooth outliers and to interpolate gaps in the data. Thus, the problem of missing values in the upper triangle is also solved in a natural way.     

An analysis of option pricing under systematic consumption risk using GARCH

An issue in the pricing of contingent claims is whether to account for consumption risk. This is relevant for contingent claims on stock indices, such as the FTSE 100 share price index, as investor’s desire for smooth consumption is often used to explain risk premiums on stock market portfolios, but is not used to explain risk premiums on contingent claims themselves. This paper addresses this fundamental question by allowing for consumption in an economy to be correlated with returns. Daily data on the FTSE 100 share price index are used to compare three option pricing models: the Black–Scholes option pricing model, a GARCH (1, 1) model priced under a risk-neutral framework, and a GARCH (1, 1) model priced under systematic consumption risk. The findings are that accounting for systematic consumption risk only provides improved accuracy for in-the-money call options. When the correlation between consumption and returns increases, the model that accounts for consumption risk will produce lower call option prices than observed prices for in-the-money call options. These results combined imply that the potential consumption-related premium in the market for contingent claims is constant in the case of FTSE 100 index options.     

Ruin probabilities and optimal capital allocation for heterogeneous life annuity portfolios

A large part of the actuarial literature is devoted to the derivation of ruin probabilities in various non-life insurance risk models. On the contrary, very few papers deal with ruin probabilities for life insurance portfolios. The difficulties arise from the dependence and non-stationarity of the annual payments made by the insurance company. This paper shows that the ruin probability in case of life annuity portfolios can be computed from algorithms derived by De Pril (1989) and Dhaene & Vandebroek (1995). Approximations for ruin probabilities are discussed. The present article complements the works of Frostig et al. (2003) who considered whole life, endowment, and temporary assurances, and of Denuit & Frostig (2008) who considered homogeneous life annuities portfolios. Here, heterogeneous portfolios (with respect to age and/or face amounts) are studied. Particular attention is paid to the capital allocation problem. The total amount of reserve is shared among the risk classes in order to minimize the ruin probability. It is then fair to charge a higher margin to the risk classes requiring more capital.     

Stochastic Payments per Claim Incurred

We propose a Bayesian model to quantify the uncertainty associated with the payments per claim incurred (PPCI) algorithm. Based on the PPCI algorithm, two submodels are proposed for the number of reported claims run-off triangle and the PPCI run-off triangle, respectively. The model for the claims amount is then derived from the two submodels under the assumption of independence between the number of incurred claims and the PPCI. The joint likelihood of the number of reported claims and claims amount is derived. The posterior distribution of parameters is estimated via the Hamiltonian Monte Carlo (HMC) sampling approach. The Bayesian estimator, the process variance, the estimation variance, and the predictive distribution of unpaid claims are also studied. The proposed model and the HMC inference engine are applied to to an empirical claims dataset of the WorkSafe Victoria to estimate the unpaid claims of the doctor benefit. The Bayesian modeling procedure is further refined by including a preliminary generalized linear model analysis. The results are compared with those in a PwC report. An alternative model is compared with the proposed model based on various information criteria.     

Lognormal Mixed Models for Reported Claims Reserves

Abstract

Traditional claims-reserving techniques are based on so-called run-off triangles containing aggregate claim figures. Such a triangle provides a summary of an underlying data set with individual claim figures. This contribution explores the interpretation of the available individual data in the framework of longitudinal data analysis. Making use of the theory of linear mixed models, a flexible model for loss reserving is built. Whereas traditional claims-reserving techniques don’t lead directly to predictions for individual claims, the mixed model enables such predictions on a sound statistical basis with, for example, confidence regions. Both a likelihood-based as well as a Bayesian approach are considered. In the frequentist approach, expressions for the mean squared error of prediction of an individual claim reserve, origin year reserves, and the total reserve are derived. Using MCMC techniques, the Bayesian approach allows simulation from the complete predictive distribution of the reserves and the calculation of various risk measures. The paper ends with an illustration of the suggested techniques on a data set from practice, consisting of Belgian automotive third-party liability claims. The results for the mixed-model analysis are compared with those obtained from traditional claims-reserving techniques for run-off triangles. For the data under consideration, the lognormal mixed model fits the observed individual data well. It leads to individual predictions comparable to those obtained by applying chain-ladder development factors to individual data. Concerning the predictive power on the aggregate level, the mixed model leads to reasonable predictions and performs comparable to and often better than the stochastic chain ladder for aggregate data.     

我国保险产品结构的合理性分析

近年来,我国保险市场存在着产品结构不合理的现实问题,虽然强调"转方式、调结构",但主要是针对寿险业务结构的调整,并未注意到寿险业务与非寿险业务比例结构的合理性,财产保险市场发展薄弱.本文首先从产品保障功能的角度对产品结构合理性的衡量问题进行了讨论,引入了"非寿险业务占比"指标作为衡量保险业发展健康程度的指标.通过对国际...     

我国产险公司利用赔款准备金实现盈余管理的实证研究

利用我国产险业2001--2005年的财务数据,本文采用盈余频率分布模型,发现产险公司存在盈余管理现象。在此基础上,本文实证分析产险公司赔款准备金与盈余管理及其他财务指标之间的关联性,揭示公司调整赔款准备金的动机。结果表明,产险公司存在利用赔款准备金评估方式修饰盈余的动机。当承保结果不佳、赔付恶化,或者投资收益较低时,产险公司会低估赔款准备金,虚增账面盈余。应对监营也是产险公司利用赔款准备金进行盈余管理的重要诱因。中、外资产险公司税赋动机存在显著的差异。2005年赔款准备金评估方法的改变并未显著减弱产险公司利用赔款准备金进行盈余管理的动机,但赔款准备金提取的谨慎程度显著提高。     

Tainted portfolios: How impairment accounting rules restrict security sales

Contrary to claims that fair value accounting exacerbated banks’ securities sales during the recent financial crisis, we present evidence that suggests – if anything – that the current impairment accounting rules served as a deterrent to selling. Specifically, because banks must provide evidence of their ‘intent and ability’ to hold securities with unrealized losses, there are strong incentives to reduce, rather than increase, security sales when market values decline to avoid ‘tainting’ their remaining securities portfolio. Validating this concern, we find that banks incur greater other‐than‐temporary impairment (OTTI) charges when they sell more securities. We then find that banks sell fewer securities when their security portfolios have larger unrealized losses (and thus larger potential impairment charges), and these results are concentrated in banks with homogenous securities portfolios, expert auditors, more experienced managers, and greater regulatory capital slack. Overall, our results suggest that – contrary to critics’ claims – the accounting rules appear to have reduced banks’ propensity to sell their securities during the financial crisis.     

Abnormal Returns from Takeover Prediction Modelling: Challenges and Suggested Investment Strategies

While takeover targets earn significant abnormal returns, studies tend to find no abnormal returns from investing in predicted takeover targets. In this study, we show that the difficulty of correctly identifying targets ex ante does not fully explain the below‐expected returns to target portfolios. Target prediction models’ inability to optimally time impending takeovers, by taking account of pre‐bid target underperformance and the anticipation of potential targets by other market participants, diminishes but does not eliminate the potential profitability of investing in predicted targets. Importantly, we find that target portfolios are predisposed to underperform, as targets and distressed firms share common firm characteristics, resulting in the misclassification of a disproportionately high number of distressed firms as potential targets. We show that this problem can be mitigated, and significant risk‐adjusted returns can be earned, by screening firms in target portfolios for size, leverage and liquidity.     

Rethinking age-period-cohort mortality trend models

Longevity risk arising from uncertain mortality improvement is one of the major risks facing annuity providers and pension funds. In this article, we show how applying trend models from non-life claims reserving to age-period-cohort mortality trends provides new insight in estimating mortality improvement and quantifying its uncertainty. Age, period and cohort trends are modelled with distinct effects for each age, calendar year and birth year in a generalised linear models framework. The effects are distinct in the sense that they are not conjoined with age coefficients, borrowing from regression terminology, we denote them as main effects. Mortality models in this framework for age-period, age-cohort and age-period-cohort effects are assessed using national population mortality data from Norway and Australia to show the relative significance of cohort effects as compared to period effects. Results are compared with the traditional Lee–Carter model. The bilinear period effect in the Lee–Carter model is shown to resemble a main cohort effect in these trend models. However, the approach avoids the limitations of the Lee–Carter model when forecasting with the age-cohort trend model.