Risks

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2741 KiB

Open AccessFeature PaperArticle

Understanding Reporting Delay in General Insurance

by Richard J. Verrall and Mario V. Wüthrich

Risks 2016, 4(3), 25; https://doi.org/10.3390/risks4030025 - 8 Jul 2016

Cited by 24 | Viewed by 5849

The aim of this paper is to understand and to model claims arrival and reporting delay in general insurance. We calibrate two real individual claims data sets to the statistical model of Jewell and Norberg. One data set considers property insurance and the [...] Read more.

The aim of this paper is to understand and to model claims arrival and reporting delay in general insurance. We calibrate two real individual claims data sets to the statistical model of Jewell and Norberg. One data set considers property insurance and the other one casualty insurance. For our analysis we slightly relax the model assumptions of Jewell allowing for non-stationarity so that the model is able to cope with trends and with seasonal patterns. The performance of our individual claims data prediction is compared to the prediction based on aggregate data using the Poisson chain-ladder method. Full article

(This article belongs to the Special Issue Non-Life Insurance Mathematics beyond Risk Theory: Pricing and Claims Reserving)

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950 KiB

Open AccessFeature PaperArticle

Ruin Probabilities with Dependence on the Number of Claims within a Fixed Time Window

by Corina Constantinescu, Suhang Dai, Weihong Ni and Zbigniew Palmowski

Risks 2016, 4(2), 17; https://doi.org/10.3390/risks4020017 - 15 Jun 2016

Cited by 13 | Viewed by 5341

Abstract

We analyse the ruin probabilities for a renewal insurance risk process with inter-arrival times depending on the claims that arrive within a fixed (past) time window. This dependence could be explained through a regenerative structure. The main inspiration of the model comes from [...] Read more.

We analyse the ruin probabilities for a renewal insurance risk process with inter-arrival times depending on the claims that arrive within a fixed (past) time window. This dependence could be explained through a regenerative structure. The main inspiration of the model comes from the bonus-malus (BM) feature of pricing car insurance. We discuss first the asymptotic results of ruin probabilities for different regimes of claim distributions. For numerical results, we recognise an embedded Markov additive process, and via an appropriate change of measure, ruin probabilities could be computed to a closed-form formulae. Additionally, we employ the importance sampling simulations to derive ruin probabilities, which further permit an in-depth analysis of a few concrete cases. Full article

(This article belongs to the Special Issue Non-Life Insurance Mathematics beyond Risk Theory: Pricing and Claims Reserving)

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1237 KiB

Open AccessArticle

Estimating Quantile Families of Loss Distributions for Non-Life Insurance Modelling via L-Moments

by Gareth W. Peters, Wilson Ye Chen and Richard H. Gerlach

Risks 2016, 4(2), 14; https://doi.org/10.3390/risks4020014 - 20 May 2016

Cited by 21 | Viewed by 5216

Abstract

This paper discusses different classes of loss models in non-life insurance settings. It then overviews the class of Tukey transform loss models that have not yet been widely considered in non-life insurance modelling, but offer opportunities to produce flexible skewness and kurtosis features [...] Read more.

This paper discusses different classes of loss models in non-life insurance settings. It then overviews the class of Tukey transform loss models that have not yet been widely considered in non-life insurance modelling, but offer opportunities to produce flexible skewness and kurtosis features often required in loss modelling. In addition, these loss models admit explicit quantile specifications which make them directly relevant for quantile based risk measure calculations. We detail various parameterisations and sub-families of the Tukey transform based models, such as the g-and-h, g-and-k and g-and-j models, including their properties of relevance to loss modelling. One of the challenges that are amenable to practitioners when fitting such models is to perform robust estimation of the model parameters. In this paper we develop a novel, efficient, and robust procedure for estimating the parameters of this family of Tukey transform models, based on L-moments. It is shown to be more efficient than the current state of the art estimation methods for such families of loss models while being simple to implement for practical purposes. Full article

(This article belongs to the Special Issue Non-Life Insurance Mathematics beyond Risk Theory: Pricing and Claims Reserving)

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438 KiB

Open AccessArticle

Macro vs. Micro Methods in Non-Life Claims Reserving (an Econometric Perspective)

by Arthur Charpentier and Mathieu Pigeon

Risks 2016, 4(2), 12; https://doi.org/10.3390/risks4020012 - 14 May 2016

Cited by 11 | Viewed by 5506

Abstract

Traditionally, actuaries have used run-off triangles to estimate reserve (“macro” models, on aggregated data). However, it is possible to model payments related to individual claims. If those models provide similar estimations, we investigate uncertainty related to reserves with “macro” and “micro” models. We [...] Read more.

Traditionally, actuaries have used run-off triangles to estimate reserve (“macro” models, on aggregated data). However, it is possible to model payments related to individual claims. If those models provide similar estimations, we investigate uncertainty related to reserves with “macro” and “micro” models. We study theoretical properties of econometric models (Gaussian, Poisson and quasi-Poisson) on individual data, and clustered data. Finally, applications in claims reserving are considered. Full article

(This article belongs to the Special Issue Non-Life Insurance Mathematics beyond Risk Theory: Pricing and Claims Reserving)

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815 KiB

Open AccessArticle

Telematics and Gender Discrimination: Some Usage-Based Evidence on Whether Men’s Risk of Accidents Differs from Women’s

by Mercedes Ayuso, Montserrat Guillen and Ana María Pérez-Marín

Risks 2016, 4(2), 10; https://doi.org/10.3390/risks4020010 - 8 Apr 2016

Cited by 52 | Viewed by 8144

Abstract

Pay-as-you-drive (PAYD), or usage-based automobile insurance (UBI), is a policy agreement tied to vehicle usage. In this paper we analyze the effect of the distance traveled on the risk of accidents among young drivers with a PAYD policy. We use regression [...] Read more.

Pay-as-you-drive (PAYD), or usage-based automobile insurance (UBI), is a policy agreement tied to vehicle usage. In this paper we analyze the effect of the distance traveled on the risk of accidents among young drivers with a PAYD policy. We use regression models for survival data to estimate how long it takes them to have their first accident at fault during the coverage period. Our empirical application with real data is presented and shows that gender differences are mainly attributable to the intensity of use. Indeed, although gender has a significant effect in explaining the time to the first crash, this effect is no longer significant when the average distance traveled per day is introduced in the model. This suggests that gender differences in the risk of accidents are, to a large extent, attributable to the fact that men drive more often than women. Estimates of the time to the first accident for different driver risk types are presented. We conclude that no gender discrimination is necessary if telematics provides enough information on driving habits. Full article

(This article belongs to the Special Issue Non-Life Insurance Mathematics beyond Risk Theory: Pricing and Claims Reserving)

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251 KiB

Open AccessArticle

Analysis of Insurance Claim Settlement Process with Markovian Arrival Processes

by Jiandong Ren

Risks 2016, 4(1), 6; https://doi.org/10.3390/risks4010006 - 11 Mar 2016

Cited by 3 | Viewed by 6070

Abstract

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, [...] Read more.

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. Full article

(This article belongs to the Special Issue Non-Life Insurance Mathematics beyond Risk Theory: Pricing and Claims Reserving)

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1223 KiB

Open AccessArticle

Multivariate Frequency-Severity Regression Models in Insurance

by Edward W. Frees, Gee Lee and Lu Yang

Risks 2016, 4(1), 4; https://doi.org/10.3390/risks4010004 - 25 Feb 2016

Cited by 85 | Viewed by 14833

Abstract

In insurance and related industries including healthcare, it is common to have several outcome measures that the analyst wishes to understand using explanatory variables. For example, in automobile insurance, an accident may result in payments for damage to one’s own vehicle, damage to [...] Read more.

In insurance and related industries including healthcare, it is common to have several outcome measures that the analyst wishes to understand using explanatory variables. For example, in automobile insurance, an accident may result in payments for damage to one’s own vehicle, damage to another party’s vehicle, or personal injury. It is also common to be interested in the frequency of accidents in addition to the severity of the claim amounts. This paper synthesizes and extends the literature on multivariate frequency-severity regression modeling with a focus on insurance industry applications. Regression models for understanding the distribution of each outcome continue to be developed yet there now exists a solid body of literature for the marginal outcomes. This paper contributes to this body of literature by focusing on the use of a copula for modeling the dependence among these outcomes; a major advantage of this tool is that it preserves the body of work established for marginal models. We illustrate this approach using data from the Wisconsin Local Government Property Insurance Fund. This fund offers insurance protection for (i) property; (ii) motor vehicle; and (iii) contractors’ equipment claims. In addition to several claim types and frequency-severity components, outcomes can be further categorized by time and space, requiring complex dependency modeling. We find significant dependencies for these data; specifically, we find that dependencies among lines are stronger than the dependencies between the frequency and average severity within each line. Full article

(This article belongs to the Special Issue Non-Life Insurance Mathematics beyond Risk Theory: Pricing and Claims Reserving)

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799 KiB

Open AccessArticle

On the Joint Analysis of the Total Discounted Payments to Policyholders and Shareholders: Dividend Barrier Strategy

by Eric C.K. Cheung, Haibo Liu and Jae-Kyung Woo

Risks 2015, 3(4), 491-514; https://doi.org/10.3390/risks3040491 - 10 Nov 2015

Cited by 11 | Viewed by 5377

Abstract

In the compound Poisson insurance risk model under a dividend barrier strategy, this paper aims to analyze jointly the aggregate discounted claim amounts until ruin and the total discounted dividends until ruin, which represent the insurer’s payments to its policyholders and shareholders, respectively. [...] Read more.

In the compound Poisson insurance risk model under a dividend barrier strategy, this paper aims to analyze jointly the aggregate discounted claim amounts until ruin and the total discounted dividends until ruin, which represent the insurer’s payments to its policyholders and shareholders, respectively. To this end, we introduce a Gerber–Shiu-type function, which further incorporates the higher moments of these two quantities. This not only unifies the individual study of various ruin-related quantities, but also allows for new measures concerning covariances to be calculated. The integro-differential equation satisfied by the generalized Gerber–Shiu function and the boundary condition are derived. In particular, when the claim severity is distributed as a combination of exponentials, explicit expressions for this Gerber–Shiu function in some special cases are given. Numerical examples involving the covariances between any two of (i) the aggregate discounted claims until ruin, (ii) the discounted dividend payments until ruin and (iii) the time of ruin are presented along with some interpretations. Full article

(This article belongs to the Special Issue Non-Life Insurance Mathematics beyond Risk Theory: Pricing and Claims Reserving)

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479 KiB

Open AccessArticle

Multi-Objective Stochastic Optimization Programs for a Non-Life Insurance Company under Solvency Constraints

by Massimiliano Kaucic and Roberto Daris

Risks 2015, 3(3), 390-419; https://doi.org/10.3390/risks3030390 - 15 Sep 2015

Cited by 6 | Viewed by 7026

Abstract

In the paper, we introduce a multi-objective scenario-based optimization approach for chance-constrained portfolio selection problems. More specifically, a modified version of the normal constraint method is implemented with a global solver in order to generate a dotted approximation of the Pareto frontier for [...] Read more.

In the paper, we introduce a multi-objective scenario-based optimization approach for chance-constrained portfolio selection problems. More specifically, a modified version of the normal constraint method is implemented with a global solver in order to generate a dotted approximation of the Pareto frontier for bi- and tri-objective programming problems. Numerical experiments are carried out on a set of portfolios to be optimized for an EU-based non-life insurance company. Both performance indicators and risk measures are managed as objectives. Results show that this procedure is effective and readily applicable to achieve suitable risk-reward tradeoff analysis. Full article

(This article belongs to the Special Issue Non-Life Insurance Mathematics beyond Risk Theory: Pricing and Claims Reserving)

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425 KiB

Open AccessArticle

Valuation of Index-Linked Cash Flows in a Heath–Jarrow–Morton Framework

by Jonas Alm and Filip Lindskog

Risks 2015, 3(3), 338-364; https://doi.org/10.3390/risks3030338 - 10 Sep 2015

Viewed by 5952

Abstract

In this paper, we study the valuation of stochastic cash flows that exhibit dependence on interest rates. We focus on insurance liability cash flows linked to an index, such as a consumer price index or wage index, where changes in the index value [...] Read more.

In this paper, we study the valuation of stochastic cash flows that exhibit dependence on interest rates. We focus on insurance liability cash flows linked to an index, such as a consumer price index or wage index, where changes in the index value can be partially understood in terms of changes in the term structure of interest rates. Insurance liability cash flows that are not explicitly linked to an index may still be valued in our framework by interpreting index returns as so-called claims inflation, i.e., an increase in claims cost per sold insurance contract. We focus primarily on the case when a deep and liquid market for index-linked contracts is absent or when the market price data are unreliable. Firstly, we present an approach for assigning a monetary value to a stochastic cash flow that does not require full knowledge of the joint dynamics of the cash flow and the term structure of interest rates. Secondly, we investigate in detail model selection, estimation and validation in a Heath–Jarrow–Morton framework. Finally, we analyze the effects of model uncertainty on the valuation of the cash flows and how forecasts of cash flows and interest rates translate into model parameters and affect the valuation. Full article

(This article belongs to the Special Issue Non-Life Insurance Mathematics beyond Risk Theory: Pricing and Claims Reserving)

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