Incurred But Not Reported Research Articles

On multivariate discounted compound renewal sums with time-dependent claims in the presence of reporting/payment delays

In this paper, we consider an insurance portfolio containing several types of policies which may simultaneously face claims arising from the same catastrophe. A renewal counting process for the number of events causing claims and multivariate claim severities which are dependent on the occurrence time and/or the delay in reporting or payment are assumed. A unified model is proposed to study the time-dependent loss quantities such as the discounted aggregate reported/unreported claims and the number of the incurred but not reported (IBNR) claims. We then derive the joint moments of (i) different types of discounted aggregate claims until time t; and (ii) different types of discounted aggregate reported/unreported claims (including the total numbers of IBNR as special case) until time t. Finally, some numerical examples involving covariances and correlations of the aforementioned quantities are provided.

A micro-level claim count model with overdispersion and reporting delays

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 so-called ‘micro-level 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 over-dispersion 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 Incurred-But-Not-Reported (IBNR) claims. The model is calibrated and illustrated using real data from the AUSI data set.

A marked Cox model for the number of IBNR claims: Theory

Incurred but not reported (IBNR) loss reserving is an important issue for Property & Casualty (P&C) insurers. To calculate IBNR reserve, one needs to model claim arrivals and then predict IBNR claims. However, factors such as temporal dependence among claim arrivals and environmental variation are often not incorporated in many of the current loss reserving models, which may greatly affect the accuracy of IBNR predictions.In this paper, we propose to model the claim arrival process together with its reporting delays as a marked Cox process. Our model is versatile in modeling temporal dependence, allowing also for natural interpretations. This paper focuses mainly on the theoretical aspects of the proposed model. We show that the associated reported claim process and IBNR claim process are both marked Cox processes with easily convertible intensity functions and marking distributions. The proposed model can also account for fluctuations in the exposure. By an order statistics property, we show that the corresponding discretely observed process preserves all the information about the claim arrivals. Finally, we derive closed-form expressions for both the autocorrelation function (ACF) and the distributions of the numbers of reported claims and IBNR claims. Model estimation and its applications are considered in a subsequent paper, Badescu et al. (2015b).

A Marked Cox Model for the Number of IBNR Claims: Estimation and Application

Incurred but not reported (IBNR) loss reserving is of great importance for Property & Casualty (P&C) insurers. However, the temporal dependence exhibited in the claim arrival process is not reflected in many current loss reserving models, which might affect the accuracy of the IBNR reserve predictions. To overcome this shortcoming, we proposed a marked Cox process and showed its many desirable properties in Badescu et al. (2015b).In this paper, we consider the model estimation and applications. We first present a generalized expectation-maximization (EM) algorithm which guarantees the efficiency of the estimators unlike the moment estimation methods widely used in estimating Cox processes. In addition, the proposed fitting algorithm can be implemented at a reasonable computational cost. We examine the performance of the proposed algorithm through simulation studies. The applicability of the proposed model is tested by fitting it to a real insurance claim data set. Through out-of-sample tests, we find that the proposed model can provide more realistic predictive distributions when compared with some traditional reserving model. Finally, we investigate and show the importance of introducing the temporal dependence in the proposed model.

Analysis of Insurance Claim Settlement Process with Markovian Arrival Processes

This paper proposes a model for the claim occurrence, reporting, and handling process of insurance companies. It is assumed that insurance claims occur according to a Markovian arrival process. An incurred claim goes through some stages of a claim reporting and handling process, such as Incurred But Not Reported (IBNR), Reported But Not Settled (RBNS) and Settled (S). We derive formulas for the joint distribution and the joint moments for the amount of INBR, RBNS and Settled claims. This model generalizes previous ones in the literature, which generally assume Poisson claim arrivals. Due to the flexibility of the Markovian arrival process, the model can be used to evaluate how the claim occurring, reporting, and handling mechanisms may affect the volatilities of the amount of IBNR, RBNS and Settled claims, and the interdependencies among them.

A Marked Cox Model for the Number of IBNR Claims: Theory

Incurred but not reported (IBNR) loss reserving is an important issue for Property & Casualty (P&C) insurers. To calculate IBNR reserve, one needs to model claim arrivals and then predict IBNR claims. However, factors such as temporal dependence among claim arrivals and environmental variation are often not incorporated in many of the current loss reserving models, which may greatly affect the accuracy of IBNR predictions.In this paper, we propose to model the claim arrival process together with its reporting delays as a marked Cox process. Our model is versatile in modeling temporal dependence, allowing also for natural interpretations. This paper focuses mainly on the theoretical aspects of the proposed model. We show that the associated reported claim process and IBNR claim process are both marked Cox processes with easily convertible intensity functions and marking distributions. The proposed model can also account for fluctuations in the exposure. By an order statistics property, we show that the corresponding discretely observed process preserves all the information about the claim arrivals. Finally, we derive closed-form expressions for both the auto-correlation function (ACF) and the distributions of the numbers of reported claims and IBNR claims. Model estimation and its applications are considered in a subsequent paper, Badescu et al. (2015b).

Modeling and predicting IBNR reserve: extended chain ladder and heteroscedastic regression analysis

This work deals with two methodologies for predicting incurred but not reported (IBNR) actuarial reserves. The first is the traditional chain ladder, which is extended for dealing with the calendar year IBNR reserve. The second is based on heteroscedastic regression models suitable to deal with the tail effect of the runoff triangle – and to forecast calendar year IBNR reserves as well. Theoretical results regarding closed expressions for IBNR predictors and mean squared errors are established – for the case of the second methodology, a Monte Carlo study is designed and implemented for accessing finite sample performances of feasible mean squared error formulae. Finally, the methods are implemented with two real data sets. The main conclusions: (i) considering tail effects does not imply theoretical and/or computational problems; and (ii) both methodologies are interesting to design softwares for IBNR reserve prediction.

A Micro-Level Claim Count Model with Overdispersion and Reporting Delays

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 so-called 'micro-level 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 over-dispersion 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 Incurred-But-Not-Reported (IBNR) claims. The model is calibrated and illustrated using real data from the AUSI data set.

Triangle-free reserving

AbstractThis paper argues that all reserving methods based on claims triangulations (the “triangle trick”), no matter how sophisticated the subsequent processing of the information contained in the triangle is, are inherently inadequate to accurately model the distribution of reserves, although they may be good enough to produce a point estimate of such reserves. The reason is that the triangle representation involves the compression (and ultimately the loss) of crucial information about the individual losses, which comes back to haunt us when we try to extract detailed information on the distribution of incurred but not reported (IBNR) and reported but not settled (RBNS) losses.This paper then argues that in order to avoid such loss of information it is necessary to adopt an approach which is similar to that used in pricing, where a separate frequency and severity model are developed and then combined by Monte Carlo simulation or other numerical techniques to produce the aggregate loss distribution.A specific implementation of this approach is described, whose core feature is a method to produce a frequency model for the incurred but not reported claim count based on the empirical distribution of delays (delay = the time between loss date and reporting date), after adjustments to make up for the bias towards smaller delays. The method also produces a kernel severity model for the individual losses, from which the severity distribution of each year of occurrence can be derived. By combining the frequency and severity model in the usual way (e.g. through Monte Carlo simulation), an aggregate model for IBNR and UPR losses can be produced.As for RBNS losses, we suggest using one of the many methods to analyse the distribution of IBNER (incurred but not enough reserved) factors to produce a possible distribution of outcomes.A case study based on real-world liability claims is used to illustrate how the method works in practice.Also, in a first step towards validating the method for calculating IBNR and comparing it with existing methods, a series of experiments with artificial data sets was undertaken, which show a drastic reduction in the prediction error of both the IBNR claim count and the IBNR total amount with respect to the standard chain ladder method. And what is perhaps most promising, the experiments show that the distribution of IBNR reserves is much closer (in terms of the Kolmogorov-Smirnov distance) to the “true” one than that based on Mack's method in the way it is normally applied. The method promises therefore a more accurate assessment of the uncertainty around reserves.

Estimating IBNR claims reserves for general insurance using Archimedean copulas

Claims reserving for general insurance business has developed significantly over the recent past. There has always been a slight mystery in short-term insurance contracts of how to go about reserving for claims, which have not yet come in, and are still in some sense of figment of the future. Insurance claims variables are non-normally distributed and therefore a measure that will capture the dependence among the variables better than the usual correlation is employed. One such method is the use of copulas. The object of this paper, therefore, is to use the Archimedean copula, Clayton copula and Frank copula to estimate outstanding incurred but not reported (IBNR) claim reserves. A comparison of the estimates of outstanding claim reserves obtained from different Archimedean copulas is also presented.

A NON-PARAMETRIC METHOD FOR INCURRED BUT NOT REPORTED CLAIM RESERVE ESTIMATION

The number and cost of claims that will arise from each policy of an insurance company’s portfolio are unknown. In fact, there is a high degree of uncertainty on how much will ultimately be the cost of claims, not only during the period of inception but also after the contract termination, since there might be future, not yet reported, losses associated with past claims. Therefore, in practice, insurance companies have to protect themselves against the possibility of this ultimate cost by creating an additional reserve known as the incurred but not reported (IBNR) reserve. This work introduces new non-parametric models to IBNR estimation based on kernel methods; namely, support vector regression and Gaussian process regression. These are used to learn certain types of nonlinear structures present in claims data using the residuals produced by a benchmark IBNR estimation model, Mack’s chain ladder. The proposed models are then compared to Mack’s model using real data examples. Our results show that the three new proposed models are competitive when compared to Mack’s benchmark model: they may produce the closest predictions of IBNR and also more accurate estimates, given that the variance for the reserve estimation, obtained through the bootstrap technique, is usually smaller than the one given by Mack’s model.

Forecasting the Pricing Kernel of IBNR Claims Development in Property-Casualty Insurance

A new method of forecasting the pricing kernel, i.e., stochastic claim inflation or link ratio function, of incurred but not reported (IBNR) claims (in property-casualty insurance) from residuals in a dynamic claims forecast model is presented. We employ a pseudo Kalman filter approach by using claims risk exposure estimates to reconstruct innovations in stochastic claims development. Whereupon we find that the pricing kernel forecast is a product measure of the innovations. We show how these results impact performance measurement including but not limited to risk-adjusted return on capital by and through insurance accounting relationships for adjusted underwriting results; and loss ratio or pure premium calculations. Additionally, we show how, in the context of Wold decomposition, diagnostics from our model can be used to compute signal to noise ratio for, and cross check, unobservable pricing kernels used to forecast claims. Furthermore, we prove that a single risk exposure factor connects seemingly unrelated specifications for loss link ratio, and claims volatility.

Stochastic modelling of insurance liabilities

Our aim is to present a method for estimating incurred but not reported (IBNR) claim reserves. Each claim is described by threecharacteristics: the claim size, the allocated loss adjusted expense and the development time. We concentrate on the joint study of all three random variables. First, the marginal univariate distributions are estimated using families of lognormal, Pareto, Wald and Gamma distributions. Next, the matrix of dependence characteristics is found between the three variables and then different multivariate copulas are used to model the joint distribution. The obtained models are fitted to the real data of motor liability insurance of a Latvian insurance company. By simulation the average claim size and allocated loss adjustment expenses in each development day have been estimated. Finally, outstanding claim reserve has been estimated.

Applying copula models to individual claim loss reserving methods

The estimation of loss reserves for incurred but not reported (IBNR) claims presents an important task for insurance companies to predict their liabilities. Recently, individual claim loss models have attracted a great deal of interest in the actuarial literature, which overcome some shortcomings of aggregated claim loss models. The dependence of the event times with the delays is a crucial issue for estimating the claim loss reserving. In this article, we propose to use semi-competing risks copula and semi-survival copula models to fit the dependence structure of the event times with delays in the individual claim loss model. A nonstandard two-step procedure is applied to our setting in which the associate parameter and one margin are estimated based on an ad hoc estimator of the other margin. The asymptotic properties of the estimators are established as well. A simulation study is carried out to evaluate the performance of the proposed methods.

A Bootstrap Estimate of the Predictive Distribution of Outstanding Claims for the Schnieper Model

AbstractThis paper considers the bootstrapping approach for measuring reserve uncertainty when applying the model of Schnieper for reserves which separate Incurred But Not Reported (IBNR) and Incurred But Not Enough Reserved (IBNER) claims. The Schnieper method has been explored in Liu and Verrall (2009), and the Mean Square Errors of Prediction (MSEP) derived. This paper takes this further by deriving the full predictive distribution, using bootstrapping. Numerical examples are provided and the MSEP from the bootstrapping approach are compared with those obtained analytically.

Semiparametric model for prediction of individual claim loss reserving

The estimation of loss reserves for incurred but not reported (IBNR) claims presents an important task for insurance companies to predict their liabilities. Conventional methods, such as ladder or separation methods based on aggregated or grouped claims of the so-called “run-off triangle”, have been illustrated to have some drawbacks. Recently, individual claim loss models have attracted a great deal of interest in actuarial literature, which can overcome the shortcomings of aggregated claim loss models. In this paper, we propose an alternative individual claim loss model, which has a semiparametric structure and can be used to fit flexibly the claim loss reserving. Local likelihood is employed to estimate the parametric and nonparametric components of the model, and their asymptotic properties are discussed. Then the prediction of the IBNR claim loss reserving is investigated. A simulation study is carried out to evaluate the performance of the proposed methods.

Some problems in actuarial finance involving sums of dependent risks

A trend in actuarial finance is to combine technical risk with interest risk. If Yt, t= 1, 2,… denotes the time–value of money (discount factors at time t) and Xt the stochastic payments to be made at time t, the random variable of interest is often the scalar product of these two random vectors V=ΣXt Yt. The vectors X? and Y? are supposed to be independent, although in general they have dependent components. The current insurance practice based on the law of large numbers disregards the stochastic financial aspects of insurance. On the other hand, introduction of the variables Y1,Y2, … to describe the financial aspects necessitates estimation or knowledge of their distribution function.We investigate some statistical models for problems of insurance and finance, including Risk Based Capital/Value at Risk, Asset Liability Management, the distribution of annuities, cash flow evaluations (in the framework of pension funds, embedded value of a portfolio, Asian options) and provisions for claims incurred, but not reported (IBNR).

Stochastic Upper Bounds for Present Value Functions

ABSTRACT In most practical cases, it is impossible to find an explicit expression for the distribution function of the present value of a sequence of cashflows that are discounted using a stochastic return process. In this article, the authors present an easily computable approximation for this distribution function. The approximation is a distribution function which is, in the sense of convex order, an upper bound for the original distribution function. Explicit examples are given for pricing stochastic annuities with a stochastic return process, for more general stochastic cash flows, as well as for pricing Asian options. Numerical results seem to indicate that the approximation will often be close to the original function. INTRODUCTION In several financial-actuarial problems, one is faced with the determination of the distribution function of random variables of the form V = [[[sigma].sup.n].sub.i=1] [[alpha].sub.i][e.sup.-[X.sub.i]] where [[alpha].sub.i] (i = 1,...,n] represents the deterministic cash flow at time i and [e.sup.-[X.sub.i]] (i = 1,...,n) is the stochastic discount factor for a payment made at time i. Hence, the random variable V can be interpreted as the present value at time 0 of a sequence of default-free payments at times 1,2,... , n. In an actuarial context, such random variables are used for describing the present value of the cash flow of an insurance portfolio [see, e.g., Dufresne (1990)]. They are also useful for the determination of incurred-but-not-reported (IBNR) reserves [see Goovaerts and Redant (1998)]. Each [[alpha].sub.i] has to be interpreted as an amount that has to be paid at time i. Equivalently, it can be said that there is an income equal to - [[alpha].sub.i] at time i. The random variable V will be called the loss variable; i.e., the present value of all future (deterministic) payments. Now assume that the distribution functions of the random variables [X.sub.i](i = 1,...,n) are known. One could assume, for example, that they are normally distributed. In reality, the random variables [X.sub.i] will certainly not be mutually independent. This means that besides the distribution functions of the [X.sub.i], the dependency structure of the random vector ([X.sub.1],...,[X.sub.n]) will also have to be taken into account in order to determine the distribution function of the loss variable V. Unfortunately, an expression for the distribution function of V is not available or is difficult to obtain in most cases. Thus, the authors shall introduce a technique that will enable bounds on V to be found. In actuarial literature, it is a common feature to replace a loss variable by a loss variable, which has a simpler structure, making it easier to determine the distribution function [see, e.g., Goovaerts, Kaas, Van Heerwaarden, and Bauwelinckx (1986)]. In order to clarify what the authors mean with a less favorable risk, the convex order will be used [see, e.g., Shaked and Shanthikumar (1994)]. Let V and W be two random variables (losses) such that E[[phi](V)] [less than or equal to] E[[phi](W)] for all convex functions [phi]: R [right arrow] R, provided the expectations exist. Then V is said to be smaller than W in the convex order (denoted as V [[less than or equal to].sub.cx] W). Roughly speaking, convex functions are functions that take on their largest values in the tails. Therefore V [[less than or equal to].sub.cx] W means that W is more likely than V to take on extreme values. Instead of saying that V is smaller than Win the convex order, it is often said that -V dominates -W in the sense of second degree stochastic dominance [see, e.g., Huang and Litzenberger (1988)]. In terms of utility theory, V [[less than or equal to].sub.cx] W means that the loss V is preferred to the loss W by all risk-averse decision makers, i.e., E[u (-V)] [greater than or equal to] E[u (-W)] for all concave utility functions u. …

IBNR reserves under stochastic interest rates

This paper intends to evaluate the present value of incurred but not reported (IBNR) reserves, when future interest rates are unknown. We first derive a result for the Laplace transform of the present value, when it is assumed that the interest rates are stochastic and can be modelled by means of a stochastic process which is similar to the model of Cox et al. (1985). Starting from this Laplace transform, it is shown how the probability distribution for the quantity under investigation can be found. The results are illustrated numerically.

An Empirical Analysis of the Magnitude and Accuracy of Incurred-But-Not-Reported Reserves

This paper investigates the Incurred-But-Not-Reported (IBNR) components of the reserves established by Property/Liability insurance companies for five liability lines. The size of the IBNR reserves relative to the size of the total reserves is measured for the time period 1975 through 1982, and the accuracy of the IBNR reserves established from 1975 to 1979 is tested. Comparisons are made between small and large firms and between stock and nonstock companies. Product mix among lines is investigated further as a possible explanatory factor of the size and accuracy of the IBNR reserves.