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Risks, Volume 4, Issue 4 (December 2016)

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Research

Open AccessArticle Optimal Reinsurance Under General Law-Invariant Convex Risk Measure and TVaR Premium Principle
Risks 2016, 4(4), 50; doi:10.3390/risks4040050
Received: 13 June 2016 / Revised: 2 December 2016 / Accepted: 9 December 2016 / Published: 16 December 2016
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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 ) = x c * + ( x - d * ) + with c * and d * being constants such that 0 c * d * . 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
Received: 15 September 2016 / Revised: 4 October 2016 / Accepted: 3 November 2016 / Published: 8 November 2016
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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 AccessArticle Bayesian Option Pricing Framework with Stochastic Volatility for FX Data
Risks 2016, 4(4), 51; doi:10.3390/risks4040051
Received: 31 August 2016 / Revised: 3 December 2016 / Accepted: 9 December 2016 / Published: 16 December 2016
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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 AccessFeature PaperArticle Incorporation of Stochastic Policyholder Behavior in Analytical Pricing of GMABs and GMDBs
Risks 2016, 4(4), 41; doi:10.3390/risks4040041
Received: 5 September 2016 / Revised: 28 October 2016 / Accepted: 1 November 2016 / Published: 8 November 2016
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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
(This article belongs to the Special Issue Ageing Population Risks)
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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
Received: 1 September 2016 / Revised: 17 October 2016 / Accepted: 3 November 2016 / Published: 9 November 2016
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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; A 1 , , A N for the claim frequencies and B 1 , , B N for the risk aversions. Depending on the distributions of A i and B i , 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 AccessArticle Multivariate TVaR-Based Risk Decomposition for Vector-Valued Portfolios
Risks 2016, 4(4), 33; doi:10.3390/risks4040033
Received: 9 June 2016 / Revised: 6 September 2016 / Accepted: 12 September 2016 / Published: 23 September 2016
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Abstract
In order to protect stakeholders of insurance companies and financial institutions against adverse outcomes of risky businesses, regulators and senior management use capital allocation techniques. For enterprise-wide risk management, it has become important to calculate the contribution of each risk within a portfolio.
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In order to protect stakeholders of insurance companies and financial institutions against adverse outcomes of risky businesses, regulators and senior management use capital allocation techniques. For enterprise-wide risk management, it has become important to calculate the contribution of each risk within a portfolio. For that purpose, bivariate lower and upper orthant tail value-at-risk can be used for capital allocation. In this paper, we present multivariate value-at-risk and tail-value-at-risk for d 2 , and we focus on three different methods to calculate optimal values for the contribution of each risk within the sums of random vectors to the overall portfolio, which could particularly apply to insurance and financial portfolios. Full article
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Open AccessArticle Parameter Estimation in Stable Law
Risks 2016, 4(4), 43; doi:10.3390/risks4040043
Received: 22 September 2016 / Revised: 7 November 2016 / Accepted: 21 November 2016 / Published: 25 November 2016
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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 AccessArticle Sharp Convex Bounds on the Aggregate Sums–An Alternative Proof
Risks 2016, 4(4), 34; doi:10.3390/risks4040034
Received: 8 May 2016 / Revised: 17 September 2016 / Accepted: 21 September 2016 / Published: 29 September 2016
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Abstract
It is well known that a random vector with given marginals is comonotonic if and only if it has the largest convex sum, and that a random vector with given marginals (under an additional condition) is mutually exclusive if and only if it
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It is well known that a random vector with given marginals is comonotonic if and only if it has the largest convex sum, and that a random vector with given marginals (under an additional condition) is mutually exclusive if and only if it has the minimal convex sum. This paper provides an alternative proof of these two results using the theories of distortion risk measure and expected utility. Full article
Open AccessArticle Estimation of Star-Shaped Distributions
Risks 2016, 4(4), 44; doi:10.3390/risks4040044
Received: 2 September 2016 / Accepted: 18 November 2016 / Published: 30 November 2016
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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 Predicting Human Mortality: Quantitative Evaluation of Four Stochastic Models
Risks 2016, 4(4), 45; doi:10.3390/risks4040045
Received: 29 August 2016 / Revised: 13 November 2016 / Accepted: 25 November 2016 / Published: 2 December 2016
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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 A Note on the Impact of Parameter Uncertainty on Barrier Derivatives
Risks 2016, 4(4), 35; doi:10.3390/risks4040035
Received: 31 August 2016 / Revised: 31 August 2016 / Accepted: 21 September 2016 / Published: 29 September 2016
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Abstract
This paper presents a comprehensive extension of pricing two-dimensional derivatives depending on two barrier constraints. We assume randomness on the covariance matrix as a way of generalizing. We analyse common barrier derivatives, enabling us to study parameter uncertainty and the risk related to
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This paper presents a comprehensive extension of pricing two-dimensional derivatives depending on two barrier constraints. We assume randomness on the covariance matrix as a way of generalizing. We analyse common barrier derivatives, enabling us to study parameter uncertainty and the risk related to the estimation procedure (estimation risk). In particular, we use the distribution of empirical parameters from IBM and EURO STOXX50. The evidence suggests that estimation risk should not be neglected in the context of multidimensional barrier derivatives, as it could cause price differences of up to 70%. Full article
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Open AccessArticle Nested MC-Based Risk Measurement of Complex Portfolios: Acceleration and Energy Efficiency
Risks 2016, 4(4), 36; doi:10.3390/risks4040036
Received: 23 June 2016 / Revised: 28 September 2016 / Accepted: 12 October 2016 / Published: 18 October 2016
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Abstract
Risk analysis and management currently have a strong presence in financial institutions, where high performance and energy efficiency are key requirements for acceleration systems, especially when it comes to intraday analysis. In this regard, we approach the estimation of the widely-employed portfolio risk
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Risk analysis and management currently have a strong presence in financial institutions, where high performance and energy efficiency are key requirements for acceleration systems, especially when it comes to intraday analysis. In this regard, we approach the estimation of the widely-employed portfolio risk metrics value-at-risk (VaR) and conditional value-at-risk (cVaR) by means of nested Monte Carlo (MC) simulations. We do so by combining theory and software/hardware implementation. This allows us for the first time to investigate their performance on heterogeneous compute systems and across different compute platforms, namely central processing unit (CPU), many integrated core (MIC) architecture XeonPhi, graphics processing unit (GPU), and field-programmable gate array (FPGA). To this end, the OpenCL framework is employed to generate portable code, and the size of the simulations is scaled in order to evaluate variations in performance. Furthermore, we assess different parallelization schemes, and the targeted platforms are evaluated and compared in terms of runtime and energy efficiency. Our implementation also allowed us to derive a new algorithmic optimization regarding the generation of the required random number sequences. Moreover, we provide specific guidelines on how to properly handle these sequences in portable code, and on how to efficiently implement nested MC-based VaR and cVaR simulations on heterogeneous compute systems. Full article
(This article belongs to the Special Issue Applying Stochastic Models in Practice: Empirics and Numerics)
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Open AccessArticle Deflation Risk and Implications for Life Insurers
Risks 2016, 4(4), 46; doi:10.3390/risks4040046
Received: 26 August 2016 / Revised: 25 November 2016 / Accepted: 28 November 2016 / Published: 3 December 2016
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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 AccessFeature PaperArticle A Note on Realistic Dividends in Actuarial Surplus Models
Risks 2016, 4(4), 37; doi:10.3390/risks4040037
Received: 3 August 2016 / Revised: 21 September 2016 / Accepted: 3 October 2016 / Published: 20 October 2016
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Abstract
Because of the profitable nature of risk businesses in the long term, de Finetti suggested that surplus models should allow for cash leakages, as otherwise the surplus would unrealistically grow (on average) to infinity. These leakages were interpreted as ‘dividends’. Subsequent literature on
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Because of the profitable nature of risk businesses in the long term, de Finetti suggested that surplus models should allow for cash leakages, as otherwise the surplus would unrealistically grow (on average) to infinity. These leakages were interpreted as ‘dividends’. Subsequent literature on actuarial surplus models with dividend distribution has mainly focussed on dividend strategies that either maximise the expected present value of dividends until ruin or lead to a probability of ruin that is less than one (see Albrecher and Thonhauser, Avanzi for reviews). An increasing number of papers are directly interested in modelling dividend policies that are consistent with actual practice in financial markets. In this short note, we review the corporate finance literature with the specific aim of fleshing out properties that dividend strategies should ideally satisfy, if one wants to model behaviour that is consistent with practice. Full article
Open AccessFeature PaperArticle Macroprudential Insurance Regulation: A Swiss Case Study
Risks 2016, 4(4), 47; doi:10.3390/risks4040047
Received: 16 September 2016 / Revised: 23 November 2016 / Accepted: 8 December 2016 / Published: 15 December 2016
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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 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
Received: 9 October 2016 / Revised: 5 December 2016 / Accepted: 7 December 2016 / Published: 16 December 2016
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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 A Note on Health Insurance under Ex Post Moral Hazard
Risks 2016, 4(4), 38; doi:10.3390/risks4040038
Received: 11 August 2016 / Revised: 12 October 2016 / Accepted: 20 October 2016 / Published: 25 October 2016
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Abstract
In the linear coinsurance problem, examined first by Mossin (1968), a higher absolute risk aversion with respect to wealth in the sense of Arrow–Pratt implies a higher optimal coinsurance rate. We show that this property does not hold for health insurance under ex
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In the linear coinsurance problem, examined first by Mossin (1968), a higher absolute risk aversion with respect to wealth in the sense of Arrow–Pratt implies a higher optimal coinsurance rate. We show that this property does not hold for health insurance under ex post moral hazard; i.e., when illness severity cannot be observed by insurers, and policyholders decide on their health expenditures. The optimal coinsurance rate trades off a risk-sharing effect and an incentive effect, both related to risk aversion. Full article
Open AccessFeature PaperArticle Compositions of Conditional Risk Measures and Solvency Capital
Risks 2016, 4(4), 49; doi:10.3390/risks4040049
Received: 14 November 2016 / Revised: 7 December 2016 / Accepted: 9 December 2016 / Published: 16 December 2016
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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 AccessFeature PaperArticle Frailty and Risk Classification for Life Annuity Portfolios
Risks 2016, 4(4), 39; doi:10.3390/risks4040039
Received: 15 September 2016 / Revised: 14 October 2016 / Accepted: 17 October 2016 / Published: 26 October 2016
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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
(This article belongs to the Special Issue Designing Post-Retirement Benefits in a Demanding Scenario)
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