Correction to: Foreword special issue Deaf 2019–Maf 2018

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The current value of the mathematical provision: a financial risk prospect

The paper addresses the question of the calculation of the current value of the mathematical provision and moulds it in a deterministic and stochastic scenario, using a proper term structure of interest rates estimated by means of a Cox-Ingersoll-Ross model. It provides a complete and original year-by-year evaluation model for the business performance, and a closed solution for the current evaluation of the reserve, together with a comprehensive insight into the dynamics of the reserve connected to the selection of a defined term structure of interest rates. Moreover, the calculation of the VaR of the mathematical provision is prospected as risk measure useful to appreciate also the evaluation rate risk. Future research prospects concern the selection of the stochastic process used to describe the dynamics of the interest rates and the possible managerial and regulatory application of a VaR measure. The modelling has been applied, as an exemplification, to a life annuity portfolio but it can be easily replicated for any kind of policy and any kind of portfolios even non homogeneous.Risk indicators, life insurance, solvency, financial risk, demographic risk

The uncertainty risk driver within a life annuity context: an overview

The paper analyzes the longevity effects on the portfolio valuations. This is a re levant topic, in particular from the perspective of insurers/sponsors of pension funds. The models chosen for actuarial calculations have to capture the survival trend and to pr oject its forecasted future impr ovements. The uncertainty in the choice is a huge concern and constitutes a relevant systematic risk driver itself, called unc ertainty risk therein. Aim of the paper is to measure the uncertainty risk and to show its trend in severa l contexts, meaningful in portfolio valuations. To this purpose the authors provide a suitable risk index and apply it in three different valuations: the initial value of an immediate life annuity portfolio; the fund of a pension annuity portfolio; the surplus of a portfolio consisting of deferred life annuities. Some graphs illustrate the results

Further Remarks on Risk Sources Measuring: The Case of a Life Annuity Portfolio

The paper considers a model that allows the actuary to measure the riskiness connected to the randomness of projected mortality tables in evaluating a portfolio of life annuities, obtaining a measure to reflect the risk associated with the randomness of the projection. The coherence of the risk parameters with the specific nature of the considered risk sources is also discussed. Numerical examples illustrate the results, showing the importance of the risk components in terms of the number of policies and comparing measure tools obtained by means of two procedures

Further Remarks on Risk Sources Measuring: The Case of a Life Annuity Portfolio

The paper considers a model that allows the actuary to measure the riskiness connected to the randomness of projected mortality tables in evaluating a portfolio of life annuities, obtaining a measure to reflect the risk associated with the randomness of the projection. The coherence of the risk parameters with the specific nature of the considered risk sources is also discussed. Numerical examples illustrate the results, showing the importance of the risk components in terms of the number of policies and comparing measure tools obtained by means of two procedures

Risk Sources in a Life Annuity Portfolio: Decomposition and Measurement Tools

The paper considers a model for a homogeneous portfolio of whole life annuities immediate. The aim is to study two risk factors: the investment risk and the insurance risk. A stochastic model of the rate of return is used to study these risk factors. Measures of the insurance risk and the investment risk for the entire portfolio are suggested. The problem of the longevity risk is presented, and its consequences with different projections of the mortality tables are analyzed. The model is applied to some concrete cases, and several illustrations show the importance of the two components of the riskiness in terms of the number of policies in the portfolio. Understanding these risks will allow insurance companies to control, to some extent, the overall risk of their annuity portfolios

The uncertainty risk driver within a life annuity context: an overview

The paper analyzes the longevity effects on the portfolio valuations. This is a relevant topic, in particular from the perspective of insurers/sponsors of pension funds. The models chosen for actuarial calculations have to capture the survival trend and to project its forecasted future improvements. The uncertainty in the choice is a huge concern and constitutes a relevant systematic risk driver itself, called uncertainty risk therein. Aim of the paper is to measure the uncertainty risk and to show its trend in several contexts, meaningful in portfolio valuations. To this purpose the authors provide a suitable risk index and apply it in three different valuations: the initial value of an immediate life annuity portfolio; the fund of a pension annuity portfolio; the surplus of a portfolio consisting of deferred life annuities. Some graphs illustrate the results

Lee–Carter model: assessing the potential to capture gender-related mortality dynamics

We investigate the ability of the Lee–Carter model to effectively estimate the gender gap ratio (GGR), the ratio between themale death rates over the female ones, by using a Cox–Ingersoll–Ross (CIR) process to provide a stochastic representation of the fitting errors. The novelty consists in the fact that we use the parameters characterizing the CIR process itself (long-term mean and volatility), in their intrinsic meanings, as quantitative measures of the long-term fitting attitude of the Lee–Carter model and synthetic indicators of the overall risk of this model. The analysis encompasses 25 European countries, to provide evidence-based indications about the goodness of fit of the Lee–Carter model in describing the GGR evolution.We highlight some stylized facts, namely systematic evidence about the fitting bias and the risk of the model across ages and countries. Furthermore, we perform a functional cluster analysis, allowing to capture similarities in the fitting performance of the Lee–Carter model among countries

Fair value and demographic aspects of the insured loans

The paper deals with the liability valuation of the insured loan in compliance of the fair value requirements for the financial assets and liabilities, as mapped out by the international boards engaged in this tool. Initially we propose a closed form for the fair valuation of the mathematical provision in a framework in which the randomness in the mortality is considered along with the financial risk component. Furthermore, the aim of the paper is to analyze the relevance of the risk arising from the demographic movements on the insured loan reserve. The approach we follow implies the mathematical provision calculated as current values, this meaning at current interest rates and at current mortality rates. In these two variables the basic risk drivers of a life insurance business dwell and the many-sided risk system consists, in its systematic aspects, in the choice of the adequate models for forecasting the future scenarios. The relevance of the impact of the risk connected to the choice of the mortality table (table risk) on the fair value of the mathematical provision is pointed out and quantified using a measurement tool obtained by conditional expectation calculus. The risk mapping is performed analyzing the accidental risk impact on the insured loan portfolio liabilities. In all likelihood, insured loan portfolios are not large enough to be considered well diversified to the aim of the pooling risk reduction; this consideration makes interesting the measuring of the liability variability caused by the random events connected to mortality (mortality risk). Practical implications of assuming different mortality scenarios on the reserve fair value are presented, a graphic description of the model risk deriving from the choice of the demographic model is provided and numerical evidences of the accidental mortality risk are show