Solvency II Directive 2009/138/EC requires an insurance and reinsurance undertakings assessment of a Solvency Capital Requirement by means of the so-called “Standard Formula” or by means of partial ...or full internal models. Focusing on the first approach, the bottom-up aggregation formula proposed by the regulator allows for a capital reduction due to the diversification effect, according to the typical subadditivity property of risk measures. However, once the overall capital has been assessed no specific allocation formula is provided or required in order to evaluate the contribution of each risk source on the overall Solvency Capital Requirement. The aim of this paper is twofold. First, we provide a closed formula for capital allocation fully compliant with the Solvency II Capital Requirement assessed by means of the Standard Formula. The solution enables a top-down approach to assess the allocated Solvency Capital Requirement among the risks considered in the Solvency II multilevel aggregation scheme; we demonstrate that the allocation formula adopted is consistent with the Euler allocation principle. Second, a solution is found as a result of an optimum capital allocation problem based on a Return On Risk Adjusted Capital measure; we establish the equivalence between the Return On Risk Adjusted Capital optimization, when the risk adjusted capital is calculated according to the Standard Formula, and the Markowitz mean-variance optimization.
The growing relevance of risk-based valuations of insurance contracts has stimulated the extension of the traditional deterministic lapse rate models towards a dynamic modelling. A popular dynamic ...model uses deterministic lapse rates as base rates and dynamic adjustment factors, generally assuming a relationship between lapses and one or more economic factors to describe policyholder behaviour. This relationship is generally represented by an S-Shaped function. This implies a monotonic increase in lapse rate by increasing the economic variable, usually set equal to a “market spread” between a benchmark rate and the policy crediting rate. In this paper, we assume a different policyholder behaviour, based on the assumption that the policyholder does not modify his/her behaviour for small values of the market spread. Hence, for a better description of such behaviour, the double-sigmoid function appears to be more adequate. The double-sigmoid function is obtained as a combination of two logits in their sum or product. Theoretical features and practical applications of the model are discussed.
This paper explores the effect of COVID-19 on health care expenditure using data from a private Health Insurance Plan (HIP). As well known, at the beginning of the COVID-19 pandemic, governments had ...to rely on Non-Pharmaceutical Interventions against the spread of the virus. However, the stringency of lockdowns differed across space and time as governments had to adjust their strategy dynamically to the country-specific development of the crisis. These strategies have strongly changed the policyholders’ behavior; however, after this period, a fundamental question is whether the policyholder behavior will return to a status quo (i.e. in traditional care delivery). We analyze these effects using a “pre-post” quantitative study using longitudinal data collected from 2017 to 2021. We consider as a consumption measure the health care expenditure amount within several types of health services, coming from a group of insured persons, followed overtime every quarter, and separating the effect per gender and age. Moving in this direction, the purpose of our contribution is to investigate if the traditional actuarial approach for assessing the loss cost, based on the Generalized Linear Models, could predict the effect on the health care expenditure due to COVID-19 and the capacity to which a HIP can anticipate these uncertainties. Our results provide a comprehensive picture of the different effects of COVID-19 on the health services offered by the HIP, as well as on the behavior ofpolicyholders during and after the pandemic period.
Abstract
The goal of this paper is the assessment of an optimal reimbursement strategy for employer-based health insurance plans (HP), that cover several categories of medical services. Indeed, a ...health plan may offer several cost-sharing provisions for these categories, and the “Percent Expense Paid” by the plan also known as “Actuarial Value” represents a summary measure of the protection provided. Starting from a wide industrial applied model in insurance for rate-making, the generalized linear models (GLM), we estimate the expected value and variance of the health expenditure for each category. Moreover, different reimbursement rules (e.g. deductibles, co-payments, policy limits, etc.) involve a change of the “Actuarial Value” calculated as the ratio between the benefits paid by the plan (i.e. reimbursement amounts) and the expenses paid by policyholders (i.e. expenditures). The latter is the percentage of expenditure reimbursed by the plan and is sometimes defined in actuarial literature as Indicated Deductible Relativity (IDR); an IDR can be calculated for each category covered by the Health Plan or per policyholder and is a commonly used method for scoring the benefits of health insurance. Hence, we calculate the optimal IDR for each category, using the optimization problem proposed by de by Finetti (Il problema dei pieni, Giornale dell’istituto italiano degli attuari, 1940) in the context of proportional reinsurance. The goal is the minimization of the variance of the total reimbursement of the Health Plan by fixing the total gain. Furthermore, we propose a numerical application to a real dataset, containing observed expenditures of an Italian HP.
This paper deals with the use of parametric quantile regression for the calculation of a loaded premium, based on a quantile measure, corresponding to individual insurance risk. Heras et al. have ...recently introduced a ratemaking process based on a two-stage quantile regression model. In the first stage, a probability to have at least one claim is estimated by a GLM logit, whereas in the second stage several quantile regressions are necessary for the estimate of the severity component. The number of quantile regressions to be performed is equal to the number of risk classes selected for ratemaking. In the actuarial context, when a large number of risk classes are considered (e.g. in Motor Third Party Liability), such approach can imply an over-parameterization and time-consuming. To this aim, in the second stage, we suggest to apply a more parsimonious approach based on Parametric Quantile Regression as introduced by Frumento and Bottai and never used in the actuarial context. This more conservative approach allows you not to lose efficiency in the estimation of premiums compared to the traditional Quantile Regression.
This article deals with the use of quantile regression and generalized linear models for a premium calculation based on quantiles. A premium principle is a functional that assigns a usually loaded ...premium to any distribution of claims. The loaded premium is generally greater than the expected value of the loss and the difference is considered to be a risk margin or a safety loading. The failure of a right charge of individual risk rate exposes the insurer to adverse selection and, consequently, to deteriorating financial results. The article aim is to define the individual pure premium rates and the corresponding risk margin in balance with some profit or solvency constraints. We develop a ratemaking process based on a two-part model using quantile regression and a gamma generalized linear model, respectively, in order to estimate the claim's severity quantiles. Generalized linear models focus on the estimation of the mean of the conditional loss distribution but have some drawbacks in assessing distribution moments other than the mean and are very sensitive to outliers. On the contrary, quantile regression exceeds these limits leading to the estimate of conditional quantiles and is more accurate for a better measurement of the variability of a risk class. The proposed methodology for premium calculation allows us to find further limits of the generalized linear model as the solution we found, under specific assumptions for a generalized linear model, is equivalent to the application of the expected value premium principle. Finally, the methodology we suggest for the two-part quantile regression allows reducing the practical issue of overmodeling and over-parameterization with respect to other proposals on the same topic.
The Italian health insurance market is currently undersized. The paucity of assured data and the discontinuous statistical surveys carried out by the National Institute of Statistics (ISTAT) ...represent one of the main obstacles to the insurance market development. The paper sets forth a parametric model to estimate technical basis for health insurance policies when data are limited and only aggregated information on mortality and morbidity is available. The probabilistic framework is based on a multiple state continuous and time inhomogeneous Markov model. We provide an estimate of transition intensities from the healthy state to the sickness state when only prevalence rates of sickness are available, according to an extension and modification of the methodology proposed in Olivieri (1996) for Long Term Care insurance. We assume that mortality intensity of both healthy and sick lives is modelled by two independent Gompertz–Makeham models.
Beta regression is a flexible tool in modeling proportions and rates, but is rarely applied in th actuarial field. In this article, we propose its application in the context of policyholder behavior ...and particularly to model surrenders and withdrawals. Surrender implies the expiration of the contract and denotes the payment of the surrender value, which is contractually defined. Withdrawal does not imply the termination of the contract and denotes the payment of a cash amount, left to the discretion of the policyholder, within the limits of the surrender value. Moreover, the Actuarial Standard of Practice 52 states that, for surrender and withdrawal estimation, the actuary should take into account several risk factors that could influence the phenomenon. To this aim, we introduce a two-part Beta regression model, where the first part consists in the estimate of the number of surrenders and withdrawals by means of a multinomial regression, as an extension of the logistic regression model frequently used in the empirical literature just to estimate surrender. Then, considering the uncertainty on the amount withdrawn, we express it as a proportion of surrender value; in this way, it assumes values continuously in the interval
and it is compliant with a Beta distribution. Therefore, in the second part, we propose the adoption of a Beta regression approach to model the proportion withdrawn of the surrender value. Our final goal is to apply our model on a real-life insurance portfolio providing the estimates of the number of surrenders and withdrawals as well as the corresponding cash amount for each risk class considered.
This paper concerns the study of the diversification effect involved in a portfolio of non-life policies priced via traditional premium principles when individual pure premiums are calculated via ...Quantile Regression. Our aim is to use Quantile Regression to estimate the individual conditional loss distribution given a vector of rating factors. To this aim, we make a comparison of the outcomes obtained via Quantile Regression with the widely used industry standard method based on generalized linear models. Then, considering a specific premium principle, we calculate individual pure premium by means of a specific functional of the conditional loss distribution, the standard deviation. We determine the portfolio risk margin according to the Solvency 2 framework and then we allocate it over each policy in a way consistent with his/her riskiness. Indeed, considering a portfolio of heterogeneous policies, we determine the individual reduction of the safety loading, due to the diversification, and we measure the risk contribution of each individual.
PDF
