Premium Calculation Research Articles

Valuation of Hybrid Financial and Actuarial Products: A Universal 3-Step Method

Financial products are priced using risk-neutral expectations justified by hedging portfolios that (as accurate as possible) match the product’s payoff. In insurance, premium calculations are based on a real-world best-estimate value plus a risk premium. The insurance risk premium is typically reduced by pooling of (in the best case) independent contracts. As hybrid life insurance contracts depend on both financial and insurance risks, their valuation requires a hybrid valuation principle that combines the two concepts of financial and actuarial valuation. The aim of this paper is to present a novel three-step projection algorithm to valuate hybrid contracts by decomposing their payoff in three parts: a financial, hedgeable part, a diversifiable actuarial part, and a residual part that is neither hedgeable nor diversifiable. The first two parts of the resulting premium are directly linked to their corresponding hedging and diversification strategies, respectively. The method allows for a separate treatment of unsystematic, diversifiable mortality risk and systematic, aggregate mortality risk related to, for example, epidemics or population-wide improvements in life expectancy. We illustrate our method in the case of CAT bonds and a pure endowment insurance contract with profit and compare the three-step method to alternative valuation operators suggested in the literature.

Consumers Perception towards Telematics in Insurance

Telematics is a branch of information technology which enables long distance transmission of computerised information from persons or objects, with the aid of equipment linked to internet. This device capture and record facts based on physical activity, and then send it to a centralized processor or server where it can be analysed and stored for the long term. Once required data has been collected by the service provider, it is examined to arrive in a conclusion. With the help of telematics devices in insurers will be in a better position to assess the risks they are underwriting. It will lead to create an improved segmentation of the market. This will allow them to increase or decrease the premium according to an individual insurance holder's risk profile, thus providing a high degree of accuracy in premium calculation and claim decisions. This paper throws light on the awareness and perception of consumers w.r.t. telematics in insurance.

Pricing the Cost of Cybercrime—A Financial Protection Approach

Infrastructures, businesses, end-users and services offered in the digitally integrated environment are exposed to a wide range of risks such as denial of service, hacking, phishing, ransomware, viruses, etc. Consequently, along with their physical life, individuals and organizations have to secure their digital life as well. Digital threats may have a major economic impact both on the individuals and the society, through the direct loss of income and/or property or even an indirect reduction of the individuals’ contribution back to the society and the state. The purpose of this paper is to study the effect of cyber-attacks to the economy, to price the associated cost and to recommend possible measures that internet service providers (ISPs) and policy makers can apply in order to mitigate these risks. In order to achieve that, we employ insurance (actuarial) pricing techniques to calculate the cost of cyber-attacks for an individual and for the economy of a country in total. We are therefore at the same time in place to recommend insurance coverage solutions that can assist in protecting the entity of interest from cyber risks. This resembles to the calculation of a risk premium, as the premium is calculated taking into account only the probability of occurrence of a cyber-attack and the interest rate and not any other loadings. In this context, we mimic the pricing of a policy that provides coverage for the cyber-attack, as well as the calculation of the amount that has to be set aside in order to compensate for the one-off economic loss suffered by the individual, as a result of its occurrence. Here lies our contribution to the scientific research in the field of cyber security insurance, as we employ insurance-based actuarial techniques in order to quantify the relevant loss.

Modeling very large losses

In this paper, we present a simple probabilistic model for aggregating very large losses into a loss collection. This supposes that “standard” losses come in various possible sizes – small, moderate and large – which, fortunately, seem to occur with decreasing frequency. Standard modeling allows us to infer a probability distribution describing their occurrence. From the historical record, we know that very large losses do occur, albeit very rarely, yet they are not usually included in the available data sets. Such losses should be made part of the distribution for computation purposes. For example, to a bank they may helpful in the computation of economic or regulatory capital, while to an insurance company they may be useful in the computation of premiums of losses due to catastrophic events. We develop a simple modeling procedure that allows us to include very large losses in a loss distribution obtained from moderately sized loss data. We say that a loss is large when it is larger than the value-at-risk (VaR) at a high confidence level. The original and extended distributions will have the same VaR but quite different values of tail VaR (TVaR).

Estimating the net premium using additional information about a quantile of the cumulative distribution function

Zhanna N. Zenkova - Associate Professor, Institute of Applied Mathematics and Computer Science, National Research Tomsk State University; SRO “Association of professional actuaries”Address: 36, Lenin Street, Tomsk, 63403047, Russian Federation E-mail: zhanna.zenkova@mail.tsu.ruElizaveta A. Krainova - Doctoral Student, Institute of Applied Mathematics and Computer Science, National Research Tomsk State University;Address: 36, Lenin Street, Tomsk, 63403047, Russian Federation E-mail: lanshakoval@gmail.com In this paper, the task of increasing the accuracy of net premium estimations in non-life insurance is considered. Improvements are achieved by involving additional information about a known q-quantile of loss cumulative distribution function. The additional information is used by projection the empirical cumulative distribution function onto the class of cumulative distribution functions with a certain q-quantile, and then the modified empirical cumulative distribution function is substituted into the integral that yields the mean value. This allows us to obtain a modified estimation of mean value using additional information about the q-quantile which is unbiased and its variance is asymptotically less than the variance of the classical sample mean, so that the mean-square error of the modification is also smaller. Therefore, the modified estimation is more accurate than the classical one for a large sample size. The influence of a quantile value on the variance of the new estimation is studied for uniform, triangular and normal distributions. It is suggested that the minimum of the variance is reached when a known quantile is equal to the median (symmetry center) for symmetrical distribution. Based on Simpson triangular distribution, it was shown that for cases of skewed distributions involving the quantile allows one to decrease the variance more significantly than for symmetrical ones. The modified estimation of mean value is applied to a real data set for calculation of a net premium. The data contain information about payments for voluntary health insurance of some insurance company. It is demonstrated that the classical method underestimates the net premium, and so it could lead to the company’s bankruptcy. After applying the new modified technique, the net premium becomes higher and the bankruptcy risk is reduced as well. This paper contains practically significant results which make it possible to give important recommendations to an insurance company.

Sample Dependence of Risk Premia

An important problem in the banking and insurance industries is that of pricing risk. The two come together when a bank buys insurance to decrease the impact of potential operational risk losses. The price of such insurance hinges on the computation of risk premiums, which involves the computation of expected values with respect to the loss distribution. When the empirical data set is not large and loss distributions are inferred from the data, a large sample dependence of the premiums on the data is to be expected. The maximum entropy-based methodologies offer model-free, non- parametric procedures to determine probability densities from empirical data with high precision. At the same time, they provide us with a framework within which to study how the sample dependence is transferred from the data to the premiums via the density. It is the aim of our paper to show how this can be done.

Good drivers pay less: A study of usage-based vehicle insurance models

Usage-based insurance (UBI) has been attracting more and more attention; however, two open research questions are how behavioral data of drivers affects driving risk and how driver behavior should affect UBI pricing schemas. This paper proposes a driver risk classification model to evaluate the risk level of drivers based on in-car sensor data. A Behavior-centric Vehicle Insurance Pricing model (BVIP) and a vehicle premium calculation prototype are developed in this paper. Based on empirical data, our research results show that BVIP achieves better accuracy in terms of risk-level classification and the prototype achieves good performance in terms of effectiveness and usability.

Crop price insurance in China: pricing and hedging using futures market

PurposeThe purpose of this paper is to demonstrate how crop price insurance premium can be calculated using an option pricing model and how insurers can transfer underwriting risks in the futures market.Design/methodology/approachBased on data from spot and futures market in China, this paper develops an improved B-S model for the calculation of crop price insurance premium and tests the possibility of hedging underwriting risks by insurance firms in the futures market.FindingsThe authors find that spot price of crops in China can be estimated with agricultural commodity futures prices, and can be taken as the insured price for crop price insurance. The authors also find that improved B-S model yields better estimation of crop price insurance premium than traditional B-S model when spot price does not follow geometric Brownian motion. Finally, the authors find that hedging can be one good alternative for insurance firms to manage underwriting risks.Originality/valueThis paper develops an improved B-S model that is data-driven in nature. Insured price of the crop price insurance, or the exercise price used in the B-S model, is estimated from a co-integration model built on spot and futures market price series. Meanwhile, distributional patterns of spot price series, one important factor determining the applicability of B-S model, is factored into the improved B-S model so that the latter is more robust and friendly to data with varied distributions. This paper also verifies the possibility of hedging of underwriting risks by insurance firms in the futures market.

Unisex pricing of German participating life annuities—Boon or bane for customer and insurance company?

This paper explores how the requirement of gender-neutral premium calculation and the new rules of surplus participation for participating life annuities (PLA) affect insurance company’s profitability and policyholder’s wellbeing. We analyse real data observed for annuity prices in the German market followed by the examination of the impact of unisex calculation, allowing for different assumed and realised gender compositions. To this end, we develop a realistically calibrated asset and liability model for a PLA with stochastic mortality, interest rates, and equity returns. We show that for males, the disadvantages of PLA’s unisex pricing as compared to the gender-specific calculation are substantially lower, if we measure them by considering a lifetime utility that accounts for both distributed surpluses and stochastic mortality. For the insurance company, the gender mix composition turns out to be less important than the capital market influence. Moreover, participating life annuities turn out to be an efficient means to share actual mortality experience between insurance company and the annuitant, and can be, at least to a certain extent, considered a substitute for indemnity longevity hedge.

Generalized Linear Mixed Models for Dependent Compound Risk Models

In ratemaking, calculation of a pure premium has traditionally been based on modeling frequency and severity in an aggregated claims model. For simplicity, it has been a standard practice to assume the independence of loss frequency and loss severity. In recent years, there is sporadic interest in the actuarial literature exploring models that departs from this independence. In this article, we extend the work of Garrido et al. (2016) which uses generalized linear models (GLMs) that account for dependence between frequency and severity and simultaneously incorporate rating factors to capture policyholder heterogeneity. In addition, we quantify and explain the contribution of the variability of claims among policyholders through the use of random effects using generalized linear mixed models (GLMMs). We calibrated our model using a portfolio of auto insurance contracts from a Singapore insurer where we observed claim counts and amounts from policyholders for a period of six years. We compared our results with the dependent GLM considered by Garrido et al. (2016), Tweedie models, and the case of independence. The dependent GLMM shows statistical evidence of positive dependence between frequency and severity. Using validation procedures, we find that the results demonstrate a more superior model when random effects are considered within a GLMM framework.

Exposure as Duration and Distance in Telematics Motor Insurance Using Generalized Additive Models

In Pay-As-You-Drive (PAYD) automobile insurance, the premium is fixed based on the distance traveled, while in usage-based insurance (UBI) the driving patterns of the policyholder are also considered. In those schemes, drivers who drive more pay a higher premium compared to those with the same characteristics who drive only occasionally, because the former are more exposed to the risk of accident. In this paper, we analyze the simultaneous effect of the distance traveled and exposure time on the risk of accident by using Generalized Additive Models (GAM). We carry out an empirical application and show that the expected number of claims (1) stabilizes once a certain number of accumulated distance-driven is reached and (2) it is not proportional to the duration of the contract, which is in contradiction to insurance practice. Finally, we propose to use a rating system that takes into account simultaneously exposure time and distance traveled in the premium calculation. We think that this is the trend the automobile insurance market is going to follow with the eruption of telematics data.

Product options for enhanced retirement income

AbstractA new stream of research proposes how people can increase their income in retirement by pooling their mortality risk. How one of these mortality risk-sharing rules could be implemented in practice, as part of a retirement income scheme, is considered. A potential advantage of the scheme is that a retiree’s housing wealth can be monetised to provide an income stream. This would mean that retirees can continue living in their home, without needing to downsize. It may be most attractive to the millions of single pensioners, particularly those who are “asset-rich and cash-poor”. Other types of assets that could be included and how to mitigate selection risks are assessed. A way of smoothing the raw mortality credits in order to make the scheme more appealing to potential members is proposed. An illustrative premium calculation suggests that the cost of the smoothing is very small compared to the potential attractiveness of an enhanced, smoothed income.

Structure Specific Flood Risk Based Insurance

This paper highlights the importance of developing accurate flood hazard maps to price insurance effectively and to communicate flood risk to interested parties. Risk-based insurance premiums can encourage insurance purchase and investment in cost effective mitigation measures. We undertake a study using light imaging detection and ranging (LIDAR) technology and depth damage curves to determine risk-based rates for residential structures in three counties in the state of North Carolina. We then compare these prices with current premiums charged to homeowners by the National Flood Insurance Program (NFIP) for 11,915 single-family residences. NFIP premiums are significantly higher than risk-based premiums for over 90 percent of the homes in each of the counties in our study. Risk-based prices are higher than NFIP premiums only in instances where buildings are predicted to suffer damage from more frequent, shallow floods that are currently not considered explicitly in NFIP premium calculations. Accurate flood maps are needed to determine cost-effective loss reduction measures and to address issues of affordability and fairness for homeowners currently living in flood-prone areas.

ON THE AGGREGATION OF EXPERTS' INFORMATION IN BONUS–MALUS SYSTEMS

AbstractIn this paper, we propose a new family of premium calculation principles based on the use of prior information from different sources. Under this framework and based on the use of Ordered Weighted Averaging operators, we provide alternative collective and Bayes premiums and describe some approaches to efficiently compute them. Several examples are detailed to illustrate the performance of the new methods.

ESTIMASI NILAI KERUGIAN DAN PREMI ASURANSI BANGUNAN AKIBAT BANJIR SUNGAI CITARUM DI KELURAHAN BALEENDAH BANDUNG

High rainfall and a lack of soil moisture absorption, often causing over flowing rivers which leads to floods. Flooding due to over flow of Citarum river is a frequent disasters in the district of Bandung which cause losses ranging from loss of property, damage to homes, diseases to loss of life. In this paper discussed about how to estimate loss cost and insurance due to flooding of the Citarum river. In this case alleged that the losses of Citarum floods affected by heads of family suffering, submerged homes, household damages and field damages. Multiple linear regression analysis was used to estimate the loss cost, which used least squares method for estimating model parameters. Mixture of analysis was used to calculate the insurance premium which considered the zone region, social class, loading factor and regulations set by OJK. The results indicate the loss cost is affected by household damage, whileother factors are not significant. Whereas, average premium is 0.5 % of insured amount by calculating the loss cost due to flooding of Citarum river. Based on estimates of loss cost and premium calculation, obtained insurance product in accordance of Baleendah condition.

Economic Valuation of Ground Mounted Photovoltaic Systems

The appraisal of ground mounted photovoltaic systems is an important question, due to increasing investments in renewable energies. The costs related to installation and maintenance, and the economic benefits related to the energy saving, suggests the use of an income approach, in order to consider the financial aspects of the photovoltaic systems. This paper proposes the use of the Discounted Cash Flow Analysis (DCFA). The DCFA allows to simulate the entire life cycle of the photovoltaic system, from the acquisition date to the end of its life cycle, to evaluate the most probable market value by discounting the annual cash flows generated by the system. In particular, the paper proposes a procedure to determine the discount rate in an innovative manner through the combination of a conventional financial method (the Build up Approach) and the analytical method which makes recourse to the use of the ascending and descending influences that act, each with positive or negative sign, on the specific risk factors related to the photovoltaic investment. To obtain an objective appraisal of the discount rate, the theory of the ascending and descending influences has been applied in this specific case for the calculation of the risk premium. The percentage incidences of the ascending and descending influences, which influence the formation of the risks to which they refer, are determined through this study for all the intrinsic factors, which are part of the photovoltaic investment risks.

Confidence intervals of the premiums of optimal bonus malus systems

In view of the economic importance of motor third-party liability insurance in developed countries the construction of optimal BMS has been given considerable interest. However, a major drawback in the construction of optimal BMS is that they fail to account for the variability on premium calculations which are treated as point estimates. The present study addresses this issue. Specifically, nonparametric mixtures of Poisson laws are used to construct an optimal BMS with a finite number of classes. The mixing distribution is estimated by nonparametric maximum likelihood (NPML). The main contribution of this paper is the use of the NPML estimator for the construction of confidence intervals for the premium rates derived by updating the posterior mean claim frequency. Furthermore, we advance one step further by improving the performance of the confidence intervals based on a bootstrap procedure where the estimated mixture is used for resampling. The construction of confidence intervals for the individual premiums based on the asymptotic maximum likelihood theory is beneficial for the insurance company as it can result in accurate and effective adjustments to the premium rating policies from a practical point of view.

Bootstrap consistency and bias correction in the nonparametric estimation of risk measures of collective risks

We consider two nonparametric estimators for the risk measure of the sum of n i.i.d. individual insurance risks divided by n, where the number of historical single claims that are used for the statistical estimation is of order n. This framework matches the situation that nonlife insurance companies are faced with within the scope of premium calculation. Indeed, the risk measure of the collective risk divided by n can be seen as a suitable premium for each of the individual risks. For both estimators asymptotic normality has been obtained recently. Here we derive almost sure bootstrap consistency for both estimators, where we allow for the weighted exchangeable bootstrap and rather general law-invariant risk measures. Both estimators are subject to a relevant negative bias for small to moderate n. For one of them we investigate by means of numerical experiments the benefit of a bootstrap-based bias correction. The numerical experiments are performed for the Value at Risk and the Average Value at Risk, and the results are comparable to those of Kim and Hardy (2007) who did analogous experiments for classical nonparametric plug-in estimators. For the other estimator the benefit of a bootstrap-based bias correction can be ruled out by theoretical arguments.

Zastosowanie modelu Bühlmanna-Strauba do estymacji stawek składki netto w systemach bonus-malus ubezpieczeń odpowiedzialności cywilnej posiadaczy pojazdów mechanicznych

One of the elements used in the process of tariff calculation of premiums in motor liability insurance is a bonus-malus system. This systems takes into account the “claims ratio” by means of increases and discounts of the base premium called net premium rates. The aim of this work is to propose an estimation method of the net premium rates in the groups of the motor third-party liability insurance portfolio of individuals. One of the maximum likelihood models, called the Buhlmann-Straub model was used for the premium estimation.

Epidemic risk and insurance coverage

AbstractIn this paper we aim to apply simple actuarial methods to build an insurance plan protecting against an epidemic risk in a population. The studied model is an extended SIR epidemic in which the removal and infection rates may depend on the number of registered removals. The costs due to the epidemic are measured through the expected epidemic size and infectivity time. The premiums received during the epidemic outbreak are measured through the expected susceptibility time. Using martingale arguments, a method by recursion is developed to calculate the cost components and the corresponding premium levels in this extended epidemic model. Some numerical examples illustrate the effect of removals and the premium calculation in an insurance plan.