Showing papers in "Scandinavian Actuarial Journal in 2017"

Characterizations of optimal reinsurance treaties: a cost-benefit approach

Abstract: This article investigates optimal reinsurance treaties minimizing an insurer’s risk-adjusted liability, which encompasses a risk margin quantified by distortion risk measures. Via the introduction of a transparent cost-benefit argument, we extend the results in Cui et al. [Cui, W., Yang, J. & Wu, L. (2013). Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles. Insurance: Mathematics and Economics 53, 74–85] and provide full characterizations on the set of optimal reinsurance treaties within the class of non-decreasing, 1-Lipschitz functions. Unlike conventional studies, our results address the issue of (non-)uniqueness of optimal solutions and indicate that ceded loss functions beyond the traditional insurance layers can be optimal in some cases. The usefulness of our novel cost-benefit approach is further demonstrated by readily solving the dual problem of minimizing the reinsurance premium while maintaining the risk-adjusted liability below a fixed tole...

Multi-population mortality models: fitting, forecasting and comparisons

Abstract: We review a number of multi-population mortality models: variations of the Li & Lee model, and the common-age-effect (CAE) model of Kleinow. Model parameters are estimated using maximum likelihood. Although this introduces some challenging identifiability problems and complicates the estimation process it allows a fair comparison of the different models. We propose to solve these identifiability problems by applying two-dimensional constraints over the parameters. Using data from six countries, we compare and rank, both visually and numerically, the models’ fitting qualities and develop forecasting models that produce non-diverging, joint mortality rate scenarios. It is found that the CAE model fits best. But we also find that the Li and Lee model potentially suffers from robustness problems when calibrated using maximum likelihood.

Ordering properties of the smallest and largest claim amounts in a general scale model

Abstract: Suppose is a set of non-negative random variables with having the distribution function , for and are independent Bernoulli random variables, independent of the ’s, with , . Let , for . It is of interest to note that in actuarial science, corresponds to the claim amount in a portfolio of risks. In this paper, under certain conditions, by using the concept of vector majorization and related orders, we discuss stochastic comparison between the smallest claim amount in the sense of the usual stochastic and hazard rate orders. We also obtain the usual stochastic order between the largest claim amounts when the matrix of parameters changes to another matrix in a mathematical sense. We then apply the results for three special cases of the scale model: generalized gamma, Marshall–Olkin extended exponential and exponentiated Weibull distributions with possibly different scale parameters to illustrate the established results.

Product pricing and solvency capital requirements for long-term care insurance

Abstract: This paper presents a comprehensive assessment of premiums, reserves and solvency capital requirements (SCRs) for long-term care (LTC) insurance policies using Activities of Daily Living and US data. We compare stand-alone policies, whole life insurance policies with LTC benefit riders (LTC insurance combined with whole life insurance), life care annuities (LTC insurance combined with annuities) and shared LTC insurance in terms of net premium cost and SCRs. Net premiums and best-estimate reserves for base LTC insurance policies are determined using Thiele’s differential equation. Product features such as the elimination period and the maximum benefit period are compared using a simulation-based model. We show how a maximum benefit period can reduce costs and risks for LTC insurance products. SCRs for longevity risk and disability risk are based on the Solvency II standard formula. We quantify the extent to which whole life insurance policies with LTC benefit riders and life care annuities provide lower S...

Lévy insurance risk process with Poissonian taxation

Abstract: The idea of taxation in risk process was first introduced by Albrecher, H. & Hipp, C. Lundberg’s risk process with tax. Blatter der DGVFM 28(1), 13–28, who suggested that a certain proportion of the insurer’s income is paid immediately as tax whenever the surplus process is at its running maximum. In this paper, a spectrally negative Levy insurance risk model under taxation is studied. Motivated by the concept of randomized observations proposed by Albrecher, H., Cheung, E.C.K. & Thonhauser, S. Randomized observation periods for the compound Poisson risk model: Dividends. ASTIN Bulletin 41(2), 645–672, we assume that the insurer’s surplus level is only observed at a sequence of Poisson arrival times, at which the event of ruin is checked and tax may be collected from the tax authority. In particular, if the observed (pre-tax) level exceeds the maximum of the previously observed (post-tax) values, then a fraction of the excess will be paid as tax. Analytic expressions for the Gerber–Shiu expected discounte...

Estimating the Gerber–Shiu function by Fourier–Sinc series expansion

Abstract: In this paper, we consider the nonparametric estimation of the Gerber–Shiu function in a compound Poisson risk model perturbed by diffusion. We present a more efficient estimator based on Fourier–Sinc series expansion. Our estimator is easily computed and has a faster convergence rate. Some simulation examples are provided to show that the estimator performs well when the sample size is finite.

A posteriori ratemaking using bivariate Poisson models

Abstract: Recently, different bivariate Poisson regression models have been used in the actuarial literature to make an a priori ratemaking taking into account the dependence between two types of claims. A natural extension for these models is to consider a posteriori ratemaking (i.e. experience rating models) that also relaxes the independence assumption. We introduce here two bivariate experience rating models that integrate the a priori ratemaking based on the bivariate Poisson regression models, extending the existing literature for the univariate case to the bivariate case. These bivariate experience rating models are applied to an automobile insurance claims data-set to analyse the consequences for posterior premiums when the independence assumption is relaxed. The main finding is that the a posteriori risk factors obtained with the bivariate experience rating models are significantly lower than those factors derived under the independence assumption.

A unifying approach to risk-measure-based optimal reinsurance problems with practical constraints

Abstract: The design of optimal reinsurance treaties in the presence of multifarious practical constraints is a substantive but underdeveloped topic in modern risk management. To examine the influence of these constraints on the contract design systematically, this article formulates a generic constrained reinsurance problem where the objective and constraint functions take the form of Lebesgue integrals whose integrands involve the unit-valued derivative of the ceded loss function to be chosen. Such a formulation provides a unifying framework to tackle a wide body of existing and novel distortion-risk-measure-based optimal reinsurance problems with constraints that reflect diverse practical considerations. Prominent examples include insurers’ budgetary, regulatory and reinsurers’ participation constraints. An elementary and intuitive solution scheme based on an extension of the cost–benefit technique in Cheung and Lo [Cheung, K.C. & Lo, A. (2015, in press). Characterizations of optimal reinsurance treaties: a cost...

A class of nonzero-sum investment and reinsurance games subject to systematic risks

Abstract: Recently, there have been numerous insightful applications of zero-sum stochastic differential games in insurance, as discussed in Liu et al. [Liu, J., Yiu, C. K.-F. & Siu, T. K. (2014). Optimal investment of an insurer with regime-switching and risk constraint. Scandinavian Actuarial Journal 2014(7), 583–601]. While there could be some practical situations under which nonzero-sum game approach is more appropriate, the development of such approach within actuarial contexts remains rare in the existing literature. In this article, we study a class of nonzero-sum reinsurance-investment stochastic differential games between two competitive insurers subject to systematic risks described by a general compound Poisson risk model. Each insurer can purchase the excess-of-loss reinsurance to mitigate both systematic and idiosyncratic jump risks of the inter-arrival claims; and can invest in one risk-free asset and one risky asset whose price dynamics follows the famous Heston stochastic volatility model [Heston, S...

Parameter risk in time-series mortality forecasts

Abstract: The projection of mortality rates is an essential part of valuing liabilities in life insurance portfolios and pension schemes. An important tool for risk management and solvency purposes is a stoc...

Optimal insurance in the presence of reinsurance

Abstract: This paper studies an optimal insurance and reinsurance design problem among three agents: policyholder, insurer, and reinsurer. We assume that the preferences of the parties are given by distortion risk measures, which are equivalent to dual utilities. By maximizing the dual utility of the insurer and jointly solving the optimal insurance and reinsurance contracts, it is found that a layering insurance is optimal, with every layer being borne by one of the three agents. We also show that reinsurance encourages more insurance, and is welfare improving for the economy. Furthermore, it is optimal for the insurer to charge the maximum acceptable insurance premium to the policyholder. This paper also considers three other variants of the optimal insurance/reinsurance models. The first two variants impose a limit on the reinsurance premium so as to prevent insurer to reinsure all its risk. An optimal solution is still layering insurance, though the insurer will have to retain higher risk. Finally, we study the...

Incorporating the Bühlmann credibility into mortality models to improve forecasting performances

Abstract: In this paper, we incorporate the Buhlmann credibility into three mortality models (the Lee–Carter model, the Cairns–Blake–Dowd model, and a linear relational model) to improve their forecasting performances, as measured by the MAPE (mean absolute percentage error), using mortality data for the UK. The results show that the MAPE reduction ratios for the three mortality models with the Buhlmann credibility are all significant. More importantly, the MAPEs under the three mortality models with the Buhlmann credibility are very close to each other for each age and forecast year. Thus, by incorporating the Buhlmann credibility we are able to converge the forecasting MAPEs resulting from the three different mortality models to a lower and more consistent level. Moreover, we provide a credibility interpretation with an individual time trend for age x and a group time trend for all ages. Finally, we apply the forecasted mortality rates both with and without the Buhlmann credibility to the net single premiums of l...

Basis Risk in Static versus Dynamic Longevity Risk Hedging

Abstract: This paper provides a tractable, parsimonious model for assessing basis risk in longevity and its effect on the hedging strategies of Pension Funds and annuity providers. Basis risk is captured by a single parameter, that measures the co-movement between the portfolio and the reference population’s longevity. The paper sets out the static, full and customized swap-hedge for an annuity, and compares it with a dynamic, partial, and index-based hedge. We calibrate our model to the UK and Scottish populations. The effectiveness of static versus dynamic strategies depends on the rebalancing frequency of the second, on the relative costs, and on basis risk, which does not affect fully-customized, static hedges. We show that appropriately calibrated dynamic hedging strategies can still be reasonably effective, even at low rebalancing frequencies.

Nonparametric estimation of the finite time ruin probability in the classical risk model

Abstract: In this paper, we consider the nonparametric estimation of the finite time ruin probability in the classical risk model. Suppose that the individual claim size distribution is unknown, but a random sample of claims is available. We construct the estimator by double Fourier transform. The asymptotic properties of the estimator are studied under large sample setting, and show that an almost convergence rate can be obtained. Some simulation examples are also provided to illustrate the performance of the estimator under finite sample size setting.

On capital injections and dividends with tax in a diffusion approximation

Abstract: We consider a diffusion approximation to a risk process with dividends and capital injections Tax has to be paid on dividends, but capital injections lead to an exemption from tax That is, tax is only paid for the aggregate excess of dividends over the capital injections The value of a strategy is the expected value of the discounted dividend payments after tax minus the discounted capital injections We solve the problem and show that the optimal dividend strategy is a barrier strategy

Forecasting disability: application of a frailty model

Abstract: Random changes in individual frailty occur due to the stochastic processes of physical deterioration or environmental influences. This paper implements a stochastic ageing model using maximum likelihood methods and calibrates the model to more than 30 years of historical Australian mortality data in order to examine cohort and gender differences in health-state distributions among older adults. We find that frailty levels have declined over time for both male and female cohorts. Nonetheless, patterns of frailty are different between genders. Older females experience a faster pace of health deterioration than their male counterparts causing them to move quicker into worse states of health. Health states are also more heterogeneous among women than men. Population-level estimates suggest that the number of elderly Australians requiring aged care services will exceed that projected under governmental assumptions by 2050.

On the relationship between classical chain ladder and granular reserving

Abstract: We connect classical chain ladder to granular reserving. This is done by defining explicitly how the classical run-off triangles are generated from individual iid observations in continuous time. One important result is that the development factors have a one to one correspondence to a histogram estimator of a hazard running in reversed development time. A second result is that chain ladder has a systematic bias if the row effect has not the same distribution when conditioned on any of the aggregated periods. This means that the chain ladder assumptions on one level of aggregation, say yearly, are different from the chain ladder assumptions when aggregated in quarters and the optimal level of aggregation is a classical bias variance trade-off depending on the data-set. We introduce smooth development factors arising from non-parametric hazard kernel smoother improving the estimation significantly.

Discrete time ruin probability with Parisian delay

Abstract: In this paper we evaluate the probability of the discrete time Parisian ruin that occurs when surplus process stays below or at zero at least for some fixed duration of time . We identify expressions for the ruin probabilities within finite and infinite-time horizon. We also find their light and heavy-tailed asymptotics when initial reserves approach infinity. Finally, we calculate these probabilities for a few explicit examples.

Analysis of IBNR claims in renewal insurance models

Abstract: Incurred but not reported (IBNR) claims, which arise naturally in insurance contexts, are of central importance to insurers for risk management and financial reporting purposes. In this paper, we first examine the moments of the total discounted IBNR claim amount at a given time when claim events occur according to a compound renewal process. Under the same claim arrival dynamic, we later consider the joint moments of the total discounted IBNR claim amount and the total incurred and reported claim amount at possibly different time points, a quantity of much interest for claim reserving purposes. In the second part of this article, we examine in more detail properties of the IBNR claim number under specific distributional assumptions for the reporting lags and the interarrival times. Among others, the self-decomposability of the IBNR claim number process is considered when claim events occur according to a compound Poisson process.

Converting retirement benefit into a life care annuity with graded benefits

Abstract: This paper deals with life care annuities, i.e. bundled products comprising a life annuity and long-term care insurance. It aims to assess the cost of converting retirement benefit into a life care annuity with graded benefits using a pre-existing public pay-as-you-go pension scheme. With this objective in mind, we present an actuarial method based on array calculus for valuing this type of life care annuity. The health dynamics of the annuitant rely on a reversible illness-death multistate framework. The paper contains a numerical example in which mortality and disability assumptions are based on data from the USA and Australia, although this should be viewed simply as an illustration. In addition, in order to check the coherence of these data, we compute life expectancy for both healthy and dependent persons, and then for dependent persons in each of the states of dependence. The effect of ruling out the recovery assumption on the annuity’s cost is also assessed. The analysis provides valuable i...

A generalization of multivariate Pareto distributions: tail risk measures, divided differences and asymptotics

Abstract: We consider a multivariate distribution of the form , where the survival function h is a multiply monotonic function of order such that , for all i and for . This generalizes work by Chiragiev and ...

CDF formulation for solving an optimal reinsurance problem

Abstract: An innovative cumulative distribution function (CDF)-based method is proposed for deriving optimal reinsurance contracts to maximize an insurer’s survival probability. The optimal reinsurance model is a non-concave constrained stochastic maximization problem, and the CDF-based method transforms it into a functional concave programming problem of determining an optimal CDF over a corresponding feasible set. Compared to the existing literature, our proposed CDF formulation provides a more transparent derivation of the optimal solutions, and more interestingly, it enables us to study a further complex model with an extra background risk and more sophisticated premium principle.

Assessing implicit hypotheses in life table construction

Abstract: Mortality figures are of capital importance for policy-making and public planning, as in forecasting financial provisions in public pension systems. General population life tables are constructed from aggregated statistics, an issue that usually entails adopting some (implicit) assumptions in their construction, such as the hypothesis of closed demographic system or the hypotheses of uniform distributions of death counts (and migration events) by age and calendar year. As microdata have become more abundant and reliable, these hypotheses could be assessed and more assumption-free estimators employed. Using a real database from Spain, we show that the above hypotheses are not appropriate in general and that the more efficient estimators proposed in this paper should be promoted, as differences persist depending on the estimator computed.

On dividends in the phase–type dual risk model

Abstract: The dual risk model assumes that the surplus of a company decreases at a constant rate over time, and grows by means of upward jumps which occur at random times with random sizes. In the present work, we study the dual risk renewal model when the waiting times are phase-type distributed. Using the roots of the fundamental and the generalized Lundberg’s equations, we get expressions for the ruin probability and the Laplace transform of the time of ruin for an arbitrary single gain distribution. Then, we address the calculation of expected discounted future dividends particularly when the individual common gains follow a phase-type distribution. We further show that the optimal dividend barrier does not depend on the initial reserve. As far as the roots of the Lundberg equations and the time of ruin are concerned, we address the existing formulae in the corresponding Sparre-Andersen insurance risk model for the first hitting time, and we generalize them to cover also the situations where we have multiple ro...

Robust bootstrap procedures for the chain-ladder method

Abstract: Insurers are faced with the challenge of estimating the future reserves needed to handle historic and outstanding claims that are not fully settled. A well-known and widely used technique is the ch...

Ruin probabilities in multivariate risk models with periodic common shock

Abstract: We propose a multidimensional risk model where the common shock affecting all classes of insurance business is arriving according to a non-homogeneous periodic Poisson process. In this multivariate setting, we derive upper bounds of Lundberg-type for the probability that ruin occurs in all classes simultaneously using the martingale approach via piecewise deterministic Markov processes theory. These results are numerically illustrated in a bivariate risk model, where the beta-shape periodic claim intensity function is considered. Under the assumption of dependent heavy-tailed claims, asymptotic bounds for the finite-time ruin probabilities associated to three types of ruin in this multivariate framework are investigated.

Integral and differential equations for the moments of multistate models in health insurance

Abstract: The moments of the random future liabilities of health insurance policies are key quantities for studying distributional properties of the future liabilities. Assuming that the randomness of the future health status of individual policyholders can be described by a semi-Markovian multistate model, integral and differential equations are derived for moments of any order and for the moment generating function. Different representations are derived and discussed with a view to numerical solution methods.

A stochastic comparison of customer classifiers with an application to customer attrition in commercial banking

Abstract: The classification of clients is an essential matter in commercial banking, insurance companies, electrical corporations, communication business, etc. Those companies frequently classify their customers by means of the information provided by the so-called classifier. Motivated by the need to compare systems of classification, we introduce a new stochastic order which permits the comparison of classifiers. The stochastic order is analysed in detail, providing characterizations and properties as well as connections with other stochastic orders and other classification systems. Such an order is applied to compare some classifiers used by a Spanish commercial banking to analyse the key problem of customer churn, obtaining conclusive results by means of real databases. Namely, the optimal classifier among them in the new stochastic order is obtained.

Pricing participating policies under the Meixner process and stochastic volatility

Abstract: We propose a model for the valuation of participating life insurance products under the Meixner process, which belongs to the family of semi-heavy tailed processes. This particular model assumption is extremely desirable as it captures the stylised features of the return distribution, with existing moment generating functions. The highlight of the paper is the analytical solution derived for minimising the relative entropy between the historical and risk-neutral measures, when driving a pricing kernel. Further, we capture the stochastic volatility effect using an accurate polynomial approximation technique. Finally, to highlight the practical applications, we conduct a simulation experiment.

Financial fairness and conditional indexation

Abstract: Collective pension contracts can generate advantages for their participants by implementing forms of risk sharing. To ensure the continuity of a collective scheme, it has to be monitored whether the contracts offered to participants are financially fair in terms of their market value. When risk sharing is implemented by means of optionalities such as conditional indexation, the analysis of financial fairness is not straightforward. In this paper, we use a stylised overlapping generations model to study financial fairness for a conditional indexation scheme. We find that financial fairness for all participants at all times is not feasible within a scheme of this type, unless the nature of indexation is such that the scheme is reduced to DC. However, financial fairness for incoming generations at the moment of entry can be realised. We show how to compute the fair contribution rate as a function of the current nominal asset/liability ratio for a given level of nominal entitlements. At low levels of the rati...