Open AccessArticle
Change Point Estimation in Panel Data without Boundary Issue
Risks 2017, 5(1), 7; doi:10.3390/risks5010007 (registering DOI) -
Abstract
Panel data of our interest consist of a moderate number of panels, while the panels contain a small number of observations. An estimator of common breaks in panel means without a boundary issue for this kind of scenario is proposed. In particular, the
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Panel data of our interest consist of a moderate number of panels, while the panels contain a small number of observations. An estimator of common breaks in panel means without a boundary issue for this kind of scenario is proposed. In particular, the novel estimator is able to detect a common break point even when the change happens immediately after the first time point or just before the last observation period. Another advantage of the elaborated change point estimator is that it results in the last observation in situations with no structural breaks. The consistency of the change point estimator in panel data is established. The results are illustrated through a simulation study. As a by-product of the developed estimation technique, a theoretical utilization for correlation structure estimation, hypothesis testing and bootstrapping in panel data is demonstrated. A practical application to non-life insurance is presented, as well. Full article
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Open AccessFeature PaperArticle
Optimal Investment and Liability Ratio Policies in a Multidimensional Regime Switching Model
Risks 2017, 5(1), 6; doi:10.3390/risks5010006 (registering DOI) -
Abstract
We consider an insurer who faces an external jump-diffusion risk that is negatively correlated with the capital returns in a multidimensional regime switching model. The insurer selects investment and liability ratio policies continuously to maximize her/his expected utility of terminal wealth. We obtain
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We consider an insurer who faces an external jump-diffusion risk that is negatively correlated with the capital returns in a multidimensional regime switching model. The insurer selects investment and liability ratio policies continuously to maximize her/his expected utility of terminal wealth. We obtain explicit solutions of optimal policies for logarithmic and power utility functions. We study the impact of the insurer’s risk aversion, the negative correlation between the external risk and the capital returns, and the regime of the economy on the optimal policy. We find, among other things, that the regime of the economy and the negative correlation between the external risk and the capital returns have a dramatic effect on the optimal policy. Full article
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Open AccessFeature PaperArticle
Minimum Protection in DC Funding Pension Plans and Margrabe Options
Risks 2017, 5(1), 5; doi:10.3390/risks5010005 -
Abstract
The regulation on the Belgian occupational pension schemes has been recently changed. The new law allows for employers to choose between two different types of guarantees to offer to their affiliates. In this paper, we address the question arising naturally: which of the
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The regulation on the Belgian occupational pension schemes has been recently changed. The new law allows for employers to choose between two different types of guarantees to offer to their affiliates. In this paper, we address the question arising naturally: which of the two guarantees is the best one? In order to answer that question, we set up a stochastic model and use financial pricing tools to compare the methods. More specifically, we link the pension liabilities to a portfolio of financial assets and compute the price of exchange options through the Margrabe formula. Full article
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Open AccessEditorial
Acknowledgement to Reviewers of Risks in 2016
Risks 2017, 5(1), 4; doi:10.3390/risks5010004 -
Abstract The editors of Risks would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2016. [...]
Full article
Open AccessFeature PaperArticle
The Effects of Largest Claim and Excess of Loss Reinsurance on a Company’s Ruin Time and Valuation
Risks 2017, 5(1), 3; doi:10.3390/risks5010003 -
Abstract
We compare two types of reinsurance: excess of loss (EOL) and largest claim reinsurance (LCR), each of which transfers the payment of part, or all, of one or more large claims from the primary insurance company (the cedant) to a reinsurer. The primary
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We compare two types of reinsurance: excess of loss (EOL) and largest claim reinsurance (LCR), each of which transfers the payment of part, or all, of one or more large claims from the primary insurance company (the cedant) to a reinsurer. The primary insurer’s point of view is documented in terms of assessment of risk and payment of reinsurance premium. A utility indifference rationale based on the expected future dividend stream is used to value the company with and without reinsurance. Assuming the classical compound Poisson risk model with choices of claim size distributions (classified as heavy, medium and light-tailed cases), simulations are used to illustrate the impact of the EOL and LCR treaties on the company’s ruin probability, ruin time and value as determined by the dividend discounting model. We find that LCR is at least as effective as EOL in averting ruin in comparable finite time horizon settings. In instances where the ruin probability for LCR is smaller than for EOL, the dividend discount model shows that the cedant is able to pay a larger portion of the dividend for LCR reinsurance than for EOL while still maintaining company value. Both methods reduce risk considerably as compared with no reinsurance, in a variety of situations, as measured by the standard deviation of the company value. A further interesting finding is that heaviness of tails alone is not necessarily the decisive factor in the possible ruin of a company; small and moderate sized claims can also play a significant role in this. Full article
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Open AccessFeature PaperArticle
On Comparison of Stochastic Reserving Methods with Bootstrapping
Risks 2017, 5(1), 2; doi:10.3390/risks5010002 -
Abstract
We consider the well-known stochastic reserve estimation methods on the basis of generalized linear models, such as the (over-dispersed) Poisson model, the gamma model and the log-normal model. For the likely variability of the claims reserve, bootstrap method is considered. In the bootstrapping
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We consider the well-known stochastic reserve estimation methods on the basis of generalized linear models, such as the (over-dispersed) Poisson model, the gamma model and the log-normal model. For the likely variability of the claims reserve, bootstrap method is considered. In the bootstrapping framework, we discuss the choice of residuals, namely the Pearson residuals, the deviance residuals and the Anscombe residuals. In addition, several possible residual adjustments are discussed and compared in a case study. We carry out a practical implementation and comparison of methods using real-life insurance data to estimate reserves and their prediction errors. We propose to consider proper scoring rules for model validation, and the assessments will be drawn from an extensive case study. Full article
Open AccessArticle
Optimal Retention Level for Infinite Time Horizons under MADM
Risks 2017, 5(1), 1; doi:10.3390/risks5010001 -
Abstract
In this paper, we approximate the aggregate claims process by using the translated gamma process under the classical risk model assumptions, and we investigate the ultimate ruin probability. We consider optimal reinsurance under the minimum ultimate ruin probability, as well as the maximum
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In this paper, we approximate the aggregate claims process by using the translated gamma process under the classical risk model assumptions, and we investigate the ultimate ruin probability. We consider optimal reinsurance under the minimum ultimate ruin probability, as well as the maximum benefit criteria: released capital, expected profit and exponential-fractional-logarithmic utility from the insurer’s point of view. Numerical examples are presented to explain how the optimal initial surplus and retention level are changed according to the individual claim amounts, loading factors and weights of the criteria. In the decision making process, we use The Analytical Hierarchy Process (AHP) and The Technique for Order of Preference by Similarity to ideal Solution (TOPSIS) methods as the Multi-Attribute Decision Making methods (MADM) and compare our results considering different combinations of loading factors for both exponential and Pareto individual claims. Full article
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Open AccessFeature PaperArticle
Compositions of Conditional Risk Measures and Solvency Capital
Risks 2016, 4(4), 49; doi:10.3390/risks4040049 -
Abstract
In this paper, we consider compositions of conditional risk measures in order to obtain time-consistent dynamic risk measures and determine the solvency capital of a life insurer selling pension liabilities or a pension fund with a single cash-flow at maturity. We first recall
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In this paper, we consider compositions of conditional risk measures in order to obtain time-consistent dynamic risk measures and determine the solvency capital of a life insurer selling pension liabilities or a pension fund with a single cash-flow at maturity. We first recall the notion of conditional, dynamic and time-consistent risk measures. We link the latter with its iterated property, which gives us a way to construct time-consistent dynamic risk measures from a backward iteration scheme with the composition of conditional risk measures. We then consider particular cases with the conditional version of the value at risk, tail value at risk and conditional expectation measures. We finally give an application of these measures with the determination of the solvency capital of a pension liability, which offers a fixed guaranteed rate without any intermediate cash-flow. We assume that the company is fully hedged against the mortality and underwriting risks. Full article
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Open AccessArticle
Bayesian Option Pricing Framework with Stochastic Volatility for FX Data
Risks 2016, 4(4), 51; doi:10.3390/risks4040051 -
Abstract
The application of stochastic volatility (SV) models in the option pricing literature usually assumes that the market has sufficient option data to calibrate the model’s risk-neutral parameters. When option data are insufficient or unavailable, market practitioners must estimate the model from the historical
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The application of stochastic volatility (SV) models in the option pricing literature usually assumes that the market has sufficient option data to calibrate the model’s risk-neutral parameters. When option data are insufficient or unavailable, market practitioners must estimate the model from the historical returns of the underlying asset and then transform the resulting model into its risk-neutral equivalent. However, the likelihood function of an SV model can only be expressed in a high-dimensional integration, which makes the estimation a highly challenging task. The Bayesian approach has been the classical way to estimate SV models under the data-generating (physical) probability measure, but the transformation from the estimated physical dynamic into its risk-neutral counterpart has not been addressed. Inspired by the generalized autoregressive conditional heteroskedasticity (GARCH) option pricing approach by Duan in 1995, we propose an SV model that enables us to simultaneously and conveniently perform Bayesian inference and transformation into risk-neutral dynamics. Our model relaxes the normality assumption on innovations of both return and volatility processes, and our empirical study shows that the estimated option prices generate realistic implied volatility smile shapes. In addition, the volatility premium is almost flat across strike prices, so adding a few option data to the historical time series of the underlying asset can greatly improve the estimation of option prices. Full article
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Open AccessArticle
Optimal Reinsurance Under General Law-Invariant Convex Risk Measure and TVaR Premium Principle
Risks 2016, 4(4), 50; doi:10.3390/risks4040050 -
Abstract
In this paper, we study the optimal reinsurance problem where risks of the insurer are measured by general law-invariant risk measures and premiums are calculated under the TVaR premium principle, which extends the work of the expected premium principle. Our objective is to
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In this paper, we study the optimal reinsurance problem where risks of the insurer are measured by general law-invariant risk measures and premiums are calculated under the TVaR premium principle, which extends the work of the expected premium principle. Our objective is to characterize the optimal reinsurance strategy which minimizes the insurer’s risk measure of its total loss. Our calculations show that the optimal reinsurance strategy is of the multi-layer form, i.e., f*(x)=xc*+(x-d*)+ with c* and d* being constants such that 0c*d*. Full article
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Open AccessFeature PaperArticle
How Does Reinsurance Create Value to an Insurer? A Cost-Benefit Analysis Incorporating Default Risk
Risks 2016, 4(4), 48; doi:10.3390/risks4040048 -
Abstract
Reinsurance is often empirically hailed as a value-adding risk management strategy which an insurer can utilize to achieve various business objectives. In the context of a distortion-risk-measure-based three-party model incorporating a policyholder, insurer and reinsurer, this article formulates explicitly the optimal insurance–reinsurance strategies
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Reinsurance is often empirically hailed as a value-adding risk management strategy which an insurer can utilize to achieve various business objectives. In the context of a distortion-risk-measure-based three-party model incorporating a policyholder, insurer and reinsurer, this article formulates explicitly the optimal insurance–reinsurance strategies from the perspective of the insurer. Our analytic solutions are complemented by intuitive but scientifically rigorous explanations on the marginal cost and benefit considerations underlying the optimal insurance–reinsurance decisions. These cost-benefit discussions not only cast light on the economic motivations for an insurer to engage in insurance with the policyholder and in reinsurance with the reinsurer, but also mathematically formalize the value created by reinsurance with respect to stabilizing the loss portfolio and enlarging the underwriting capacity of an insurer. Our model also allows for the reinsurer’s failure to deliver on its promised indemnity when the regulatory capital of the reinsurer is depleted by the reinsured loss. The reduction in the benefits of reinsurance to the insurer as a result of the reinsurer’s default is quantified, and its influence on the optimal insurance–reinsurance policies analyzed. Full article
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Open AccessFeature PaperArticle
Macroprudential Insurance Regulation: A Swiss Case Study
Risks 2016, 4(4), 47; doi:10.3390/risks4040047 -
Abstract
This article provides a case study that analyzes national macroprudential insurance regulation in Switzerland. We consider an insurance market that is based on data from the Swiss private insurance industry. We stress this market with several scenarios related to financial and insurance risks,
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This article provides a case study that analyzes national macroprudential insurance regulation in Switzerland. We consider an insurance market that is based on data from the Swiss private insurance industry. We stress this market with several scenarios related to financial and insurance risks, and we analyze the resulting risk capitals of the insurance companies. This stress-test analysis provides insights into the vulnerability of the Swiss private insurance sector to different risks and shocks. Full article
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Open AccessArticle
Deflation Risk and Implications for Life Insurers
Risks 2016, 4(4), 46; doi:10.3390/risks4040046 -
Abstract
Life insurers are exposed to deflation risk: falling prices could lead to insufficient investment returns, and inflation-indexed protections could make insurers vulnerable to deflation. In this spirit, this paper proposes a market-based methodology for measuring deflation risk based on a discrete framework: the
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Life insurers are exposed to deflation risk: falling prices could lead to insufficient investment returns, and inflation-indexed protections could make insurers vulnerable to deflation. In this spirit, this paper proposes a market-based methodology for measuring deflation risk based on a discrete framework: the latter accounts for the real interest rate, the inflation index level, its conditional variance, and the expected inflation rate. US inflation data are then used to estimate the model and show the importance of deflation risk. Specifically, the distribution of a fictitious life insurer’s future payments is investigated. We find that the proposed inflation model yields higher risk measures than the ones obtained using competing models, stressing the need for dynamic and market-consistent inflation modelling in the life insurance industry. Full article
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Open AccessArticle
Predicting Human Mortality: Quantitative Evaluation of Four Stochastic Models
Risks 2016, 4(4), 45; doi:10.3390/risks4040045 -
Abstract
In this paper, we quantitatively compare the forecasts from four different mortality models. We consider one discrete-time model proposed by Lee and Carter (1992) and three continuous-time models: the Wills and Sherris (2011) model, the Feller process and the Ornstein-Uhlenbeck (OU) process. The
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In this paper, we quantitatively compare the forecasts from four different mortality models. We consider one discrete-time model proposed by Lee and Carter (1992) and three continuous-time models: the Wills and Sherris (2011) model, the Feller process and the Ornstein-Uhlenbeck (OU) process. The first two models estimate the whole surface of mortality simultaneously, while in the latter two, each generation is modelled and calibrated separately. We calibrate the models to UK and Australian population data. We find that all the models show relatively similar absolute total error for a given dataset, except the Lee-Carter model, whose performance differs significantly. To evaluate the forecasting performance we therefore look at two alternative measures: the relative error between the forecasted and the actual mortality rates and the percentage of actual mortality rates which fall within a prediction interval. In terms of the prediction intervals, the results are more divergent since each model implies a different structure for the variance of mortality rates. According to our experiments, the Wills and Sherris model produces superior results in terms of the prediction intervals. However, in terms of the mean absolute error, the OU and the Feller processes perform better. The forecasting performance of the Lee Carter model is mostly dependent on the choice of the dataset. Full article
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Open AccessArticle
Estimation of Star-Shaped Distributions
Risks 2016, 4(4), 44; doi:10.3390/risks4040044 -
Abstract
Scatter plots of multivariate data sets motivate modeling of star-shaped distributions beyond elliptically contoured ones. We study properties of estimators for the density generator function, the star-generalized radius distribution and the density in a star-shaped distribution model. For the generator function and the
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Scatter plots of multivariate data sets motivate modeling of star-shaped distributions beyond elliptically contoured ones. We study properties of estimators for the density generator function, the star-generalized radius distribution and the density in a star-shaped distribution model. For the generator function and the star-generalized radius density, we consider a non-parametric kernel-type estimator. This estimator is combined with a parametric estimator for the contours which are assumed to follow a parametric model. Therefore, the semiparametric procedure features the flexibility of nonparametric estimators and the simple estimation and interpretation of parametric estimators. Alternatively, we consider pure parametric estimators for the density. For the semiparametric density estimator, we prove rates of uniform, almost sure convergence which coincide with the corresponding rates of one-dimensional kernel density estimators when excluding the center of the distribution. We show that the standardized density estimator is asymptotically normally distributed. Moreover, the almost sure convergence rate of the estimated distribution function of the star-generalized radius is derived. A particular new two-dimensional distribution class is adapted here to agricultural and financial data sets. Full article
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Open AccessArticle
Parameter Estimation in Stable Law
Risks 2016, 4(4), 43; doi:10.3390/risks4040043 -
Abstract For general stable distribution, cumulant function based parameter estimators are proposed. Extensive simulation experiments are carried out to validate the effectiveness of the estimates over the entire parameter space. An application to non-life insurance losses distribution is made. Full article
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Open AccessFeature PaperArticle
Optimal Premium as a Function of the Deductible: Customer Analysis and Portfolio Characteristics
Risks 2016, 4(4), 42; doi:10.3390/risks4040042 -
Abstract
An insurance company offers an insurance contract (p,K), consisting of a premium p and a deductible K. In this paper, we consider the problem of choosing the premium optimally as a function of the deductible. The insurance
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An insurance company offers an insurance contract (p,K), consisting of a premium p and a deductible K. In this paper, we consider the problem of choosing the premium optimally as a function of the deductible. The insurance company is facing a market of N customers, each characterized by their personal claim frequency, α, and risk aversion, β. When a customer is offered an insurance contract, she/he will, based on these characteristics, choose whether or not to insure. The decision process of the customer is analyzed in detail. Since the customer characteristics are unknown to the company, it models them as i.i.d. random variables; A1,,AN for the claim frequencies and B1,,BN for the risk aversions. Depending on the distributions of Ai and Bi, expressions for the portfolio size n(p;K)[0,N] and average claim frequency α(p;K) in the portfolio are obtained. Knowing these, the company can choose the premium optimally, mainly by minimizing the ruin probability. Full article
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Open AccessFeature PaperArticle
A Note on Upper Tail Behavior of Liouville Copulas
Risks 2016, 4(4), 40; doi:10.3390/risks4040040 -
Abstract
The family of Liouville copulas is defined as the survival copulas of multivariate Liouville distributions, and it covers the Archimedean copulas constructed by Williamson’s d-transform. Liouville copulas provide a very wide range of dependence ranging from positive to negative dependence in the
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The family of Liouville copulas is defined as the survival copulas of multivariate Liouville distributions, and it covers the Archimedean copulas constructed by Williamson’s d-transform. Liouville copulas provide a very wide range of dependence ranging from positive to negative dependence in the upper tails, and they can be useful in modeling tail risks. In this article, we study the upper tail behavior of Liouville copulas through their upper tail orders. Tail orders of a more general scale mixture model that covers Liouville distributions is first derived, and then tail order functions and tail order density functions of Liouville copulas are derived. Concrete examples are given after the main results. Full article
Open AccessFeature PaperArticle
Incorporation of Stochastic Policyholder Behavior in Analytical Pricing of GMABs and GMDBs
Risks 2016, 4(4), 41; doi:10.3390/risks4040041 -
Abstract
Variable annuities represent certain unit-linked life insurance products offering different types of protection commonly referred to as guaranteed minimum benefits (GMXBs). They are designed for the increasing demand of the customers for private pension provision. In this paper we analytically price variable annuities
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Variable annuities represent certain unit-linked life insurance products offering different types of protection commonly referred to as guaranteed minimum benefits (GMXBs). They are designed for the increasing demand of the customers for private pension provision. In this paper we analytically price variable annuities with guaranteed minimum repayments at maturity and in case of the insured’s death. If the contract is prematurely surrendered, the policyholder is entitled to the current value of the fund account reduced by the prevailing surrender fee. The financial market and the mortality model are affine linear. For the surrender model, a Cox process is deployed whose intensity is given by a deterministic function (s-curve) with stochastic inputs from the financial market. So, the policyholders’ surrender behavior depends on the performance of the financial market and is stochastic. The presented pricing scheme incorporates the stochastic surrender behavior of the policyholders and is only based on suitable closed-form approximations. Full article
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Open AccessFeature PaperArticle
Frailty and Risk Classification for Life Annuity Portfolios
Risks 2016, 4(4), 39; doi:10.3390/risks4040039 -
Abstract
Life annuities are attractive mainly for healthy people. In order to expand their business, in recent years, some insurers have started offering higher annuity rates to those whose health conditions are critical. Life annuity portfolios are then supposed to become larger and more
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Life annuities are attractive mainly for healthy people. In order to expand their business, in recent years, some insurers have started offering higher annuity rates to those whose health conditions are critical. Life annuity portfolios are then supposed to become larger and more heterogeneous. With respect to the insurer’s risk profile, there is a trade-off between portfolio size and heterogeneity that we intend to investigate. In performing this, there is a second and possibly more important issue that we address. In actuarial practice, the different mortality levels of the several risk classes are obtained by applying adjustment coefficients to population mortality rates. Such a choice is not supported by a rigorous model. On the other hand, the heterogeneity of a population with respect to mortality can formally be described with a frailty model. We suggest adopting a frailty model for risk classification. We identify risk groups (or classes) within the population by assigning specific ranges of values to the frailty within each group. The different levels of mortality of the various groups are based on the conditional probability distributions of the frailty. Annuity rates for each class then can be easily justified, and a comprehensive investigation of insurer’s liabilities can be performed. Full article
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