In this paper models for claimfrequency and claim size in non-life insurance are con- sidered. 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 com- pound Poisson model going back to Lundberg (1903), we allow for dependencies between claim size and claimfrequency. Both models for the individual and average claim sizes of a policyholder are considered. A fully Bayesian approach is followed, parameters are estimated using Markov Chain Monte Carlo (MCMC). The issue of model comparison is thoroughly ad- dressed. Besides the deviance information criterion suggested by Spiegelhalter et al. (2002), the predictive model choice criterion (Gelfand and Ghosh (1998)) and proper scoring rules (Gneiting and Raftery (2005)) based on the posterior predictive distribution are investigated. 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 claimfrequency and claim size and also leads to more accurate predictions of the total claim sizes. Further we quantify the significant number of claims effects on claim size.
First, we detected overdispersion in the Poisson model and employed the negative-binomial model to show that considering heterogeneity in insurance policies yields better ﬁ t of the model, as well as more precise estimates of claimfrequency. Second, we analysed the linear eﬀ ect of continuous rating factors. We showed that using non-linear systematic component also yield the better ﬁ t of the model, as well as generate new hypotheses to be veriﬁ ed further, especially the stability of the fractional polynomial functions. Finally, using interaction between owner’s age and gender, we demonstrated that considering interactions in the model may also increase the ﬁ t of the model and generate new hypotheses to be analysed further, i.e. the ﬁ ndings that the eﬀ ect of gender is changing dependently on the owner’s age.
The smoothed quantile function (3.1), its properties, and probabilistic behavior of its estimator are valid for discrete distributions with ﬁnite support, i.e., when d < ∞ . For discrete distributions with inﬁnite support, such as Poisson, negative binomial or their zero-inﬂated versions, d = ∞ . Since such distributions are essential for modeling claimfrequency, we need to extend the results of Chapter 3 to the case d = ∞ . Thus, in this chapter we ﬁrst investigate the proposal of Wang and Hutson (2011) on how to deal with such distributions, then make a new proposal and evaluate its performance, theoretically and via simulations.
Thus, in this research, we focus on the MOD insurance in the Russian motor insurance market. As it was shown above, the prominent increase in the prices of policies in the MOD insurance market was influenced by a number of factors, including changes in the CMPTL insurance market (although the CMTPL prices themselves are rigid and strictly regulated by the Central Bank of Russia). Nevertheless, we share the opinion of some experts in the Russian insurance business that the best way to avoid MOD policies over-pricing is a more accurate risk assessment and taking into account an extended range of possible factors influencing claimfrequency and claim severity in Russian MOD insurance market.
In today’s society, we place a high importance on modelling and predicting various types of risks. This allows for protection against various financial insecurities that might otherwise cause significant harm to our financial security. Such risks we model include non-life insurance or property and casualty insurance. In the field of personal casualty insurance, actuaries are often tasked with modelling auto insurance claims. The goal of the insurance company is to calculate an effective insurance price or premium to the corresponding insured party in order to cover the necessary risk. Claimfrequency, also known as count data, and claim severity are the variables used to calculate the average cost of claims for property and casualty insurance. A superior model for claimfrequency and claim severity means more competitive fees and in turn, a more profitable coverage for the insurer. Therefore, modelling claimfrequency and claim severity is a crucial step for pricing personal and casualty insurance.
Exhibit 4 displays changes in lost-time claimfrequency by Size of Loss. Each claim cost represents undeveloped paid losses plus case reserves as of 1st report. For this snapshot, we did not account for medical or wage inflation. Hence, a migration from the low to high ranges is evident. For example, a $48,000 claim in 2004 would fall in the $10K to $50K range. A comparable claim in 2008 would likely cost more than $50,000, just due to inflation, and would, therefore, appear in the next higher size of loss range ($50K to $250K).
In Exhibit 13, we have assigned all lost-time claims into one of two categories (Likely-to-Develop and Not-Likely-to-Develop) based on Part of Body. NCCI identifies Likely-to-Develop claims as those with body parts such as head, skull, neck, trunk, spinal cord, upper and lower back, or multiple body parts. 15 Not-Likely-to-Develop claims include those involving fingers, hand, arm, wrist, toes, foot, and ankle. Likely-to-Develop claims typically have higher loss development than Not-Likely-to- Develop claims beyond 1st report. Likely-to-Develop claims experienced a sharper decline in claimfrequency per payroll from PYE 2008 to 2012.
The 2SLS results from estimating equation (1) are displayed in Table 8. OLS results are presented in column (1) of this table for the purpose of comparing with previous literature. Without accounting for policy endogeneity, damage caps have a negative and statistically signiﬁcant effect on claimfrequency. If noneconomic damage caps reduce the incentives to ﬁle by reducing the expected beneﬁts of ﬁling a claim, then we might expect to ﬁnd this result. Using the OLS estimates, one would conclude that a cap reduces claim fre- quency by about 11%. When the instrument variables procedure is used, how- ever, the coefﬁcient is no longer statistically signiﬁcant. Columns (2) and (3) in Table 8 employ the predicted cumulative probabilities obtained from the logit model as the instrument of choice. Table 8 also displays the conﬁdence interval around Cap. As seen in column (2), this coefﬁcient is imprecise. 32 The interval ranges from a 32% decrease to a 21% increase in suits per 100,000 of the pop- ulation. It is important to note that while the standard errors are slightly larger in the 2SLS case, the lack of statistical signiﬁcance on the variable Cap is not the sole result of substantially larger 2SLS standard errors. The magnitude of the point estimate is also dramatically reduced as well.
The third factor reviewed had, by far, the largest impact on the 2010 change in frequency. The denominator of the NCCI accident year frequency measure is calendar year earned premium, which includes audit premiums related to prior years’ policies. Audits booked (and earned) in 2010 were significantly lower than anticipated as a result of the recession. Under more stable economic conditions, premium audits typically produce additions to premium. However, during the recession, it became apparent that estimated payrolls overstated final payroll, and, therefore, audits resulted in return premiums. This change in the direction of premium audits had a significant impact on the calendar year earned premiums used in the denominator of the NCCI accident year frequency calculation. NCCI estimates that the Calendar Year 2010 premium understated the premium on actual exposures earned by 3%. In contrast, the Calendar Year 2009 premium overstated the premium on actual exposures earned by 2%. These distortions combined to produce a five-percentage-point overstatement in the claimfrequency change for 2010, as measured using calendar year earned premium.
Actuarial modelling in property insurance may be broken down on claim size (next chapter) and claimfrequency (treated here). Section 3.2 introduced the Poisson distribution which was used as model for the numbers of claims. The parameter was λ = µT (for single policies) and λ = JµT (for portfolios) where J was the number of policies, µ claim intensity and T the time of exposure. Virtually all claim number models used in practice is related to the Poisson distribution in some way. Such a line has strong theoretical support through the Poisson point process outlined in Section 8.2. This leads to the Poisson model as a plausible one under a wide range of circumstances. It also explores the meaning of the key parameter µ, the vehicle for model extensions. There are here two main viewpoints. The first (with a long tradition in actuarial science) is to regard µ as a random variable, either re-drawn independently for each customer or re-drawn each period as common background for all. Models of that kind were initiated in Section 6.3, and there will be more below. Then there are situations where variations in µ are linked to explanatory factors, such as young drivers causing accidents more often than old or eartquackes or hurricanes striking certaing regions more frequently than others. In a similar vein risk may be growing systematically over time or being influenced by the season of the year, as in Figure 8.1 below. Such situations are best treated through Poisson regression, introduced in Section 8.4.
The best evidence that physicians’ behavior can be altered by reducing the frequency with which plaintiffs sue, or the amounts that can be re- covered when they do, comes from a study of the impact of malpractice risk on Caesarean delivery rates in New York State ( 128, 129). That study, which found a systematic relationship between the strength of various malpractice risk measures (i.e., claimfrequency and insurance premiums) and Caesarean delivery rates, is consistent with the hypothesis that tort reforms that reduce claimfrequency or malpractice premiums will reduce defensive behavior. Yet. it is unknown how far Localio’s findings for obstetricians and Caesarean rates can be generalized to other states, specialties. clinical situations, or procedures-especially in light of the failure of other studies funded by OTA to find a correlation between malpractice risk and clinical behavior.
For this argument to be of practical use, the adequacy of these induced models should be comparable to that of the separate GLMs for frequency and severity. Now, these induced models have an important limitation, that makes this unlikely: under the Tweedie GLM, a larger pure premium implies both, a larger claimfrequency and claim severity. To see why this is the case, let µ i be the mean of the i-th class of a Tweedie GLM, that means
Actuaries in insurance companies try to design a tariff structure that will fairly distribute the burden of claims among policyholders. Therefore they try to find the best model for an estimation of the insurance premium. The paper deals with an estimate of a priori annual claimfrequency and application of bonus-malus system in the vehicle insurance. In this paper, analysis of the portfolio of vehicle insurance data using generalized linear model (GLM) is performed. Based on large real-world sample of data from 67 857 vehicles, the present study proposes a classification analysis approach addressing the selection of predictor variables. The models with different predictor variables are compared by the analysis of deviance. Based on this comparison, the model for the best estimate of annual claimfrequency is chosen. Then the bonus-malus (BM) system is used for each class of drivers and Bayesian relative premium is calculated. Finally a fairer premium for different groups of drivers is proposed.
The estimated claimfrequency was also affected by the frequency model used. However, the effect of the model complexity consisted of increased differentiation of expected frequencies across policies rather than significant differences in SCR estimates. In other words, if we respected non-linearity or interactions, we did not obtain significantly different SCR estimates. Furthermore, these frequency models were more important for the individual claim estimates and setting the premium. By comparing the SCR estimates determined by the frequency-severity models, we identified that neglecting the heterogeneity in claimfrequency (i.e. using the Poisson model) was as consequential as neglecting the occurrence of large claims. Finally, the approach combining the mixed model and any negative- binomial model yielded the highest SCR estimates.
To construct a tariff structure that reflects the various risk profiles in a portfolio within a reasonable and statistically sound basis, actuaries usually rely on regres- sion techniques. Such techniques allow for the inclusion of various explanatory (also called classifying or rating) variables so that the actuary is able to construct risk classes with more or less similar risk profiles. For non-life (also called: property and casualty or general) insurance, typical response variables in these regression models are the number of claims (or claimfrequency) per unit of exposure on the one hand, and the corresponding amount of loss, given a claim (or claim severity) on the other hand. A formal discussion of these actuarial terminologies will follow in Sect. 2.1. That section also explains how regression models for claims frequency and severity allow us to estimate the price of risk.
Running a car is expensive enough without paying more than you have to for your car insurance. That’s why earning a no claims discount could make paying for a premium a little easier on your pocket. Who knows how much you could save on your car insurance premiums by aiming for a year’s claim free driving.
They will prevent further damage and their scope of work doesn’t have to be approved by the insurance company. Next, call your insurance carrier, report the claim, and get a claim number. They will get a field adjuster out to assess the damage.
constitutes an agreement by the claimant to assign to the Fund any rights, claims, and causes of action the claimant has against any person for the costs and damages which are the subject of the compensated claims and to cooperate reasonably with the Fund in any claim or action by the Fund against any person to recover the amounts paid by the Fund. The cooperation shall include, but is not limited to, immediately reimbursing the Fund for any compensation received from any other source for the same costs and damages and providing any documentation, evidence, testimony, and other support, as may be necessary for the Fund to recover from any person. 33 CFR § 136.115(a).
The scheme operates under ‘no-fault’ legislation. This means that an injured employee does not have to prove negligence on the part of his or her employer for his or her claim to be successful. For a guide on how Comcare determines claims made under the Safety, Rehabilitation and Compensation Act 1988 (SRC Act) visit http://www.comcare.gov.au.